What was the River Ister in the Time of Strabo? A Mathematical Approach

Abstract

We introduce a novel method for map registration and apply it to transformation of the river Ister from Strabo’s map of the World to the current map in the World Geodetic System. This transformation leads to the surprising but convincing result that Strabo’s river Ister best coincides with the nowadays Tauernbach-Isel-Drava-Danube course and not with the Danube river what is commonly assumed. Such a result is supported by carefully designed mathematical measurements and it resolves all related controversies otherwise appearing in understanding and translation of Strabo’s original text. Based on this result, we also show that Strabo’s Suevi in the Hercynian Forest corresponds to the Slavic people in the Carpathian-Alpine basin and thus that the compact Slavic settlement was there already at the beginning of the first millennium AD.

Full text

Στράβωνος Γεωγραφικά

῎Ιστρος

῎Ιστρος ῥέων πρὸς νότον κατ᾿ ἀρχάς εἶτ᾿ ἐπιστρέφων εὐθὺς ἀπὸ
τῆς δύσεως ἐπὶ τὴν ἀνατολὴν καὶ τὸν Πόντον ἄρχεται μὲν οὖν ἀπὸ
τῶν Γερμανικῶν ἄκρων τῶν ἑσπερίων πλησίον δὲ καὶ τοῦ μυχοῦ τοῦ
᾿Αδριατικοῦ διέχων αὐτοῦ περὶ χιλίους σταδίους
χιλίους σταδίους

ἄκρων
γνησίους
γνήσιοι γὰρ οἱ Γερμανοὶ κατὰ τὴν ῾Ρωμαίων διάλεκτον
῾Ρῆνος

Σαβάτα
῎Οκρᾳ ῎Αλπεις
῎Αλπεια ῎Αλβια

᾿Ισάρας ῎Αταγι ς
῎Ιστρος

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y=Ax+b,
x=(x1,x
2)M1y=(y1,y
2)
M2
A=a1a2
a3a4
2×2
b=b1
b2
y1=a1x1+a2x2+b1,
y2=a3x1+a4x2+b2.
Ab
|y−(Ax+b)|2
x∈M1y∈M2
x1,...,xny1,...,yn
n
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Tpkpk,k =1,2,3
k=1
k=2 k=3
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pj={xi,yi;i=1,...,n
pj}
εpj
mean (Tpk)=1
npj
xi∈pj
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DEyi,Tpk(xi)yi
Tpk(xi)
εpj
max (Tpk)= max
xi∈pj
DEyi,Tpk(xi).

Tpk
pj
pkεp1
mean (Tpk)εp2
mean (Tpk)εp3
mean (Tpk)
7.527 256.856 723.619
197.655 24.259 126.697
155.672 189.790 22.981
Tpk
pj
pkεp1
max (Tpk)εp2
max (Tpk)εp3
max (Tpk)
12.201 383.310 917.010
235.104 32.042 178.451
198.112 232.691 35.989
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TpG
pk,k =1,2,3
TpG
pG
pkεpk
mean (TpG)εpk
max (TpG)
45.730 65.453
62.224 92.509
43.519 79.303
52.543 92.509
pk
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TpG
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mean (TpG)εpk
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p1
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Ω
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≈ui−1,j −2ui,j +ui+1,j +ui,j−1−2ui,j +ui,j+1
=ui−1,j +ui+1,j −4ui,j +ui,j−1+ui,j+1,
ui,j uxi,j

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ui−1,j =ui+1,j
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=2ui+1,j −4ui,j +ui,j−1+ui,j +1
xi,j ∈E(pk):ui,j =v(pk)
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xi,j ∈ E(pk)∧i=N1,j =N2:−2ui−1,j −2ui,j−1+4ui,j =0
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xi,j ∈ E(pk)∧i=2,...,N
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xi,j ∈ E(pk)∧i=1,j =2,...,N
2−1:−ui,j−1+4ui,j −2ui+1,j −ui,j+1 =0
pkM1
pk
M1
M1M2

Ω A −J
E(pk)
pk
C−J∂Ω
p1p2p3
Tp1Tp2Tp3

῎Ιστρος
Λούγδουνον

῾Ροδανὸς
᾿Αδούλας
Λημέννα λίμνη
•
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A={a1,...,anA}.
ai,i=1,...,n
A

A={
a1,...,
anA}

a1=a1,a1+a2
2,

ai=ai−1+ai
2,ai∪ai,ai+ai+1
2,i=2,...,n
A−1,

anA=anA−1+anA
2,anA,
u,v u v ai=|
ai|
LA
A
δH(A, B)
A={a1,...,anA}B={b1,...,bnB}

A={
a1,...,
anA}
B=
b1,...,
bnB

δH(A, B)= 1
LA
nA

i=1aimin

bj∈

B
DE(ai,
bj),
DE(ai,
bj)ai
bj∈
B
H(A, B)
H(A, B)= LAδH(A, B)+LBδH(B, A)
LA+LB
.
δH(A, B)
δH(A, B)= sup
ai∈A
inf

bj∈

B
DEai,
bj
H(A, B)
H(A, B)=max{δH(A, B),δ
H(B, A)}.
ai∈
A
dtB
Lm(A, B, dt)
Lm(A, B, dt)= 

ai∈

Aai,
min

bj∈

B
DE(ai,
bj)<d
t.
x=1,2,3

300 km

ABδH(A, B)H(A, B)δH(A, B )H(A, B)
317.781 112.802
206.829 79.573
154.611 44.903
149.693 39.621
154.611 44.717
149.693 39.484
317.781 177.243
206.829 130.218
39.020 21.831
38.809 20.547
39.020 21.383
40.628 20.173
154.611 65.344
149.693 59.130

Lm(A, B, dt)
LAdt=10 dt=50 dt= 100
228.127 (8.4 ) 633.263 (23.3 ) 1114.154 (41.0 )
185.952 (13.0 ) 493.414 (34.7 ) 782.470 (55.0 )
207.040 (10.0 ) 563.339 (27.2 ) 948.312 (45.8 )
299.190 (14.7 ) 1285.049 (63.5 ) 1693.837 (83.7 )
267.783 (18.8 ) 1011.383 (71.1 ) 1247.252 (87.7 )
283.486 (16.4 ) 1148.216 (66.6 ) 1470.545 (85.4 )
320.930 (15.8 ) 1289.600 (63.6 ) 1698.389 (83.8 )
287.667 (20.2 ) 1011.383 (71.1 ) 1247.252 (87.7 )
304.299 (17.6 ) 1150.492 (66.7 ) 1472.821 (85.4 )
16.857 (1.1 ) 85.824 (5.9 ) 157.926 (11.0 )
24.146 (3.9 ) 99.998 (16.1 ) 153.185 (24.7 )
20.501 (2.0 ) 92.911 (9.0 ) 155.555 (15.1 )
87.920 (11.9 ) 737.609 (100.0 ) 737.609 (100.0 )
105.976 (17.1 ) 617.966 (100.0 ) 617.966 (100.0 )
96.948 (14.3 ) 677.788 (100.0 ) 677.788 (100.0 )
109.660 (14.7 ) 742.161 (100.0 ) 742.161 (100.0 )
125.860 (20.3 ) 617.966 (100.0 ) 617.966 (100.0 )
117.760 (17.3 ) 680.063 (100.0 ) 680.063 (100.0 )
211.270 (16.4 ) 547.439 (42.6 ) 956.228 (74.4 )
161.806 (20.1 ) 393.416 (48.9 ) 629.285 (78.3 )
186.538 (17.8 ) 470.428 (45.0 ) 792.757 (75.9 )

ABδH(A, B)H(A, B)δH(A, B )H(A, B)
278.962 116.695
194.446 92.584
198.848 66.131
189.915 57.998
198.848 65.625
189.915 56.683
278.962 157.633
194.446 125.898
54.656 35.270
66.735 35.623
54.656 33.857
70.863 32.194
198.848 96.661
189.915 88.555

Lm(A, B, dt)
LAdt=10 dt=50 dt= 100
61.260 (2.2 ) 481.784 (17.7 ) 948.962 (34.9 )
55.447 (3.8 ) 323.687 (22.4 ) 641.770 (44.5 )
58.353 (2.8 ) 402.736 (19.3 ) 795.366 (38.2 )
26.097 (1.2 ) 1082.523 (53.5 ) 1508.358 (74.5 )
26.768 (1.8 ) 852.692 (59.1 ) 1144.834 (79.4 )
26.433 (1.5 ) 967.608 (55.8 ) 1326.596 (76.6 )
41.406 (2.0 ) 1087.075 (53.6 ) 1512.910 (74.6 )
50.989 (3.5 ) 855.911 (59.3 ) 1144.834 (79.4 )
46.197 (2.6 ) 971.493 (56.0 ) 1328.872 (76.6 )
34.437 (2.4 ) 112.681 (7.8 ) 178.212 (12.4 )
25.465 (3.9 ) 90.754 (14.0 ) 143.805 (22.2 )
29.951 (2.8 ) 101.717 (9.7 ) 161.008 (15.5 )
0.000 (0.0 ) 713.419 (96.7 ) 737.609 (100.0 )
0.000 (0.0 ) 619.759 (95.8 ) 646.869 (100.0 )
0.000 (0.0 ) 666.589 (96.2 ) 692.239 (100.0 )
15.308 (2.0 ) 717.971 (96.7 ) 742.161 (100.0 )
24.221 (3.7 ) 622.978 (96.3 ) 646.869 (100.0 )
19.764 (2.8 ) 670.474 (96.5 ) 694.515 (100.0 )
26.097 (2.0 ) 369.103 (28.7 ) 770.749 (60.0 )
26.768 (3.3 ) 232.933 (29.3 ) 497.964 (62.6 )
26.433 (2.5 ) 301.018 (28.9 ) 634.357 (61.0 )

ABδH(A, B)H(A, B)δH(A, B )H(A, B)
296.586 117.455
205.818 92.198
219.255 84.164
205.818 74.903
219.255 83.610
205.818 74.175
296.586 145.940
194.015 115.984
101.197 65.823
94.308 60.811
101.197 64.181
94.308 58.773
219.255 112.879
205.818 99.975

Lm(A, B, dt)
LAdt=10 dt=50 dt= 100
46.134 (1.6 ) 327.386 (12.0 ) 911.606 (33.5 )
40.573 (2.7 ) 248.261 (16.9 ) 674.137 (45.9 )
43.354 (2.0 ) 287.824 (13.7 ) 792.871 (37.9 )
22.798 (1.1 ) 374.882 (18.5 ) 1390.515 (68.7 )
20.561 (1.4 ) 301.285 (20.5 ) 1130.088 (76.9 )
21.679 (1.2 ) 338.083 (19.3 ) 1260.301 (72.2 )
48.427 (2.3 ) 379.434 (18.7 ) 1395.067 (68.8 )
48.975 (3.3 ) 296.829 (20.2 ) 1130.088 (76.9 )
48.701 (2.7 ) 338.131 (19.3 ) 1262.577 (72.2 )
19.581 (1.3 ) 122.200 (8.5 ) 257.106 (17.9 )
10.404 (1.7 ) 57.143 (9.5 ) 144.355 (24.0 )
14.992 (1.4 ) 89.671 (8.8 ) 200.730 (19.7 )
0.000 (0.0 ) 191.060 (25.9 ) 727.479 (98.6 )
0.000 (0.0 ) 178.483 (29.7 ) 600.305 (100.0 )
0.000 (0.0 ) 184.771 (27.6 ) 663.892 (99.2 )
25.628 (3.4 ) 195.612 (26.3 ) 732.031 (98.6 )
28.413 (4.7 ) 174.027 (28.9 ) 600.305 (100.0 )
27.021 (4.0 ) 184.819 (27.5 ) 666.168 (99.2 )
22.798 (1.7 ) 183.821 (14.3 ) 652.906 (50.8 )
20.561 (2.3 ) 122.801 (14.1 ) 529.782 (61.0 )
21.679 (2.0 ) 153.311 (14.2 ) 591.344 (54.9 )

῾Ερκυνίου δρυμος
Σοήβων
ὅπου αἱ τοῦ ῎Ιστρου πηγαὶ πλησίον Σοήβων καὶ τοῦ ῾Ερκυνίου δρυμοῦ

ὥστ᾿ ἀνάγκη τῷ ἐκ τῆς Κελτικῆς ἐπὶ τὸν ῾Ερκύνιον δρυμὸν ἰόντι
πρῶτον μὲν διαπερᾶσαι τὴν λίμνην ἔπειτα τὸν ῎Ιστρον εἶτ᾿ ἤδη δι᾿
εὐπετεστέρων χωρίων ἐπὶ τὸν δρυμὸν τὰς προβάσεις ποιεῖσθαι δι᾿ ὀροπε-
δίων
ἔστι δὲ καὶ ἄλλη ὕλη μεγάλη Γαβρῆτα ἐπὶ τάδε τῶν Σοήβων ἐπέκεινα
δ᾿ ὁ ῾Ερκύνιος δρυμός ἔχεται δὲ κἀκεῖνος ὑπ᾿ αὐτῶν
Γαβρῆτα
οἱ τοίνυν ῞Ελληνες τοὺς Γέτας Θρᾷκας ὑπελάμβανον

εἶτ᾿ εὐθὺς ἡ τῶν Γετῶν συνάπτει γῆ κατ᾿ ἀρχὰς μὲν στενή παρατε-
ταμένη τῷ ῎Ιστρῳ κατὰ τὸ νότιον μέρος κατὰ δὲ τοὐναντίον τῇ παρωρείᾳ
τοῦ ῾Ερκυνίου δρυμοῦ

137 ×1.62 km = 222,24

Δίερνα

ΔΔίερνα

+