A history of Greek mathematics Vol.II from Aristarchus to Diophantus by Heath, Thomas Little, Sir, 1921
MACEDONIA is GREECE and will always be GREECE- (if they are desperate to steal a name, Monkeydonkeys suits them just fine) ΚΑΤΩ Η ΣΥΓΚΥΒΕΡΝΗΣΗ ΤΩΝ ΠΡΟΔΟΤΩΝ!!! ΦΕΚ,ΚΚΕ,ΚΝΕ,ΚΟΜΜΟΥΝΙΣΜΟΣ,ΣΥΡΙΖΑ,ΠΑΣΟΚ,ΝΕΑ ΔΗΜΟΚΡΑΤΙΑ,ΕΓΚΛΗΜΑΤΑ,ΔΑΠ-ΝΔΦΚ, MACEDONIA,ΣΥΜΜΟΡΙΤΟΠΟΛΕΜΟΣ,ΠΡΟΣΦΟΡΕΣ,ΥΠΟΥΡΓΕΙΟ,ΕΝΟΠΛΕΣ ΔΥΝΑΜΕΙΣ,ΣΤΡΑΤΟΣ, ΑΕΡΟΠΟΡΙΑ,ΑΣΤΥΝΟΜΙΑ,ΔΗΜΑΡΧΕΙΟ,ΝΟΜΑΡΧΙΑ,ΠΑΝΕΠΙΣΤΗΜΙΟ,ΛΟΓΟΤΕΧΝΙΑ,ΔΗΜΟΣ,LIFO,ΛΑΡΙΣΑ, ΠΕΡΙΦΕΡΕΙΑ,ΕΚΚΛΗΣΙΑ,ΟΝΝΕΔ,ΜΟΝΗ,ΠΑΤΡΙΑΡΧΕΙΟ,ΜΕΣΗ ΕΚΠΑΙΔΕΥΣΗ,ΙΑΤΡΙΚΗ,ΟΛΜΕ,ΑΕΚ,ΠΑΟΚ,ΦΙΛΟΛΟΓΙΚΑ,ΝΟΜΟΘΕΣΙΑ,ΔΙΚΗΓΟΡΙΚΟΣ,ΕΠΙΠΛΟ, ΣΥΜΒΟΛΑΙΟΓΡΑΦΙΚΟΣ,ΕΛΛΗΝΙΚΑ,ΜΑΘΗΜΑΤΙΚΑ,ΝΕΟΛΑΙΑ,ΟΙΚΟΝΟΜΙΚΑ,ΙΣΤΟΡΙΑ,ΙΣΤΟΡΙΚΑ,ΑΥΓΗ,ΤΑ ΝΕΑ,ΕΘΝΟΣ,ΣΟΣΙΑΛΙΣΜΟΣ,LEFT,ΕΦΗΜΕΡΙΔΑ,ΚΟΚΚΙΝΟ,ATHENS VOICE,ΧΡΗΜΑ,ΟΙΚΟΝΟΜΙΑ,ΕΝΕΡΓΕΙΑ, ΡΑΤΣΙΣΜΟΣ,ΠΡΟΣΦΥΓΕΣ,GREECE,ΚΟΣΜΟΣ,ΜΑΓΕΙΡΙΚΗ,ΣΥΝΤΑΓΕΣ,ΕΛΛΗΝΙΣΜΟΣ,ΕΛΛΑΔΑ, ΕΜΦΥΛΙΟΣ,ΤΗΛΕΟΡΑΣΗ,ΕΓΚΥΚΛΙΟΣ,ΡΑΔΙΟΦΩΝΟ,ΓΥΜΝΑΣΤΙΚΗ,ΑΓΡΟΤΙΚΗ,ΟΛΥΜΠΙΑΚΟΣ, ΜΥΤΙΛΗΝΗ,ΧΙΟΣ,ΣΑΜΟΣ,ΠΑΤΡΙΔΑ,ΒΙΒΛΙΟ,ΕΡΕΥΝΑ,ΠΟΛΙΤΙΚΗ,ΚΥΝΗΓΕΤΙΚΑ,ΚΥΝΗΓΙ,ΘΡΙΛΕΡ, ΠΕΡΙΟΔΙΚΟ,ΤΕΥΧΟΣ,ΜΥΘΙΣΤΟΡΗΜΑ,ΑΔΩΝΙΣ ΓΕΩΡΓΙΑΔΗΣ,GEORGIADIS,ΦΑΝΤΑΣΤΙΚΕΣ ΙΣΤΟΡΙΕΣ, ΑΣΤΥΝΟΜΙΚΑ,ΦΙΛΟΣΟΦΙΚΗ,ΦΙΛΟΣΟΦΙΚΑ,ΙΚΕΑ,ΜΑΚΕΔΟΝΙΑ,ΑΤΤΙΚΗ,ΘΡΑΚΗ,ΘΕΣΣΑΛΟΝΙΚΗ,ΠΑΤΡΑ, ΙΟΝΙΟ,ΚΕΡΚΥΡΑ,ΚΩΣ,ΡΟΔΟΣ,ΚΑΒΑΛΑ,ΜΟΔΑ,ΔΡΑΜΑ,ΣΕΡΡΕΣ,ΕΥΡΥΤΑΝΙΑ,ΠΑΡΓΑ,ΚΕΦΑΛΟΝΙΑ, ΙΩΑΝΝΙΝΑ,ΛΕΥΚΑΔΑ,ΣΠΑΡΤΗ,ΠΑΞΟΙ
MACEDONIA is GREECE and will always be GREECE- (if they are desperate to steal a name, Monkeydonkeys suits them just fine)
ΚΑΤΩ Η ΣΥΓΚΥΒΕΡΝΗΣΗ ΤΩΝ ΠΡΟΔΟΤΩΝ!!!
ΦΕΚ,ΚΚΕ,ΚΝΕ,ΚΟΜΜΟΥΝΙΣΜΟΣ,ΣΥΡΙΖΑ,ΠΑΣΟΚ,ΝΕΑ ΔΗΜΟΚΡΑΤΙΑ,ΕΓΚΛΗΜΑΤΑ,ΔΑΠ-ΝΔΦΚ, MACEDONIA,ΣΥΜΜΟΡΙΤΟΠΟΛΕΜΟΣ,ΠΡΟΣΦΟΡΕΣ,ΥΠΟΥΡΓΕΙΟ,ΕΝΟΠΛΕΣ ΔΥΝΑΜΕΙΣ,ΣΤΡΑΤΟΣ, ΑΕΡΟΠΟΡΙΑ,ΑΣΤΥΝΟΜΙΑ,ΔΗΜΑΡΧΕΙΟ,ΝΟΜΑΡΧΙΑ,ΠΑΝΕΠΙΣΤΗΜΙΟ,ΛΟΓΟΤΕΧΝΙΑ,ΔΗΜΟΣ,LIFO,ΛΑΡΙΣΑ, ΠΕΡΙΦΕΡΕΙΑ,ΕΚΚΛΗΣΙΑ,ΟΝΝΕΔ,ΜΟΝΗ,ΠΑΤΡΙΑΡΧΕΙΟ,ΜΕΣΗ ΕΚΠΑΙΔΕΥΣΗ,ΙΑΤΡΙΚΗ,ΟΛΜΕ,ΑΕΚ,ΠΑΟΚ,ΦΙΛΟΛΟΓΙΚΑ,ΝΟΜΟΘΕΣΙΑ,ΔΙΚΗΓΟΡΙΚΟΣ,ΕΠΙΠΛΟ, ΣΥΜΒΟΛΑΙΟΓΡΑΦΙΚΟΣ,ΕΛΛΗΝΙΚΑ,ΜΑΘΗΜΑΤΙΚΑ,ΝΕΟΛΑΙΑ,ΟΙΚΟΝΟΜΙΚΑ,ΙΣΤΟΡΙΑ,ΙΣΤΟΡΙΚΑ,ΑΥΓΗ,ΤΑ ΝΕΑ,ΕΘΝΟΣ,ΣΟΣΙΑΛΙΣΜΟΣ,LEFT,ΕΦΗΜΕΡΙΔΑ,ΚΟΚΚΙΝΟ,ATHENS VOICE,ΧΡΗΜΑ,ΟΙΚΟΝΟΜΙΑ,ΕΝΕΡΓΕΙΑ, ΡΑΤΣΙΣΜΟΣ,ΠΡΟΣΦΥΓΕΣ,GREECE,ΚΟΣΜΟΣ,ΜΑΓΕΙΡΙΚΗ,ΣΥΝΤΑΓΕΣ,ΕΛΛΗΝΙΣΜΟΣ,ΕΛΛΑΔΑ, ΕΜΦΥΛΙΟΣ,ΤΗΛΕΟΡΑΣΗ,ΕΓΚΥΚΛΙΟΣ,ΡΑΔΙΟΦΩΝΟ,ΓΥΜΝΑΣΤΙΚΗ,ΑΓΡΟΤΙΚΗ,ΟΛΥΜΠΙΑΚΟΣ, ΜΥΤΙΛΗΝΗ,ΧΙΟΣ,ΣΑΜΟΣ,ΠΑΤΡΙΔΑ,ΒΙΒΛΙΟ,ΕΡΕΥΝΑ,ΠΟΛΙΤΙΚΗ,ΚΥΝΗΓΕΤΙΚΑ,ΚΥΝΗΓΙ,ΘΡΙΛΕΡ, ΠΕΡΙΟΔΙΚΟ,ΤΕΥΧΟΣ,ΜΥΘΙΣΤΟΡΗΜΑ,ΑΔΩΝΙΣ ΓΕΩΡΓΙΑΔΗΣ,GEORGIADIS,ΦΑΝΤΑΣΤΙΚΕΣ ΙΣΤΟΡΙΕΣ, ΑΣΤΥΝΟΜΙΚΑ,ΦΙΛΟΣΟΦΙΚΗ,ΦΙΛΟΣΟΦΙΚΑ,ΙΚΕΑ,ΜΑΚΕΔΟΝΙΑ,ΑΤΤΙΚΗ,ΘΡΑΚΗ,ΘΕΣΣΑΛΟΝΙΚΗ,ΠΑΤΡΑ, ΙΟΝΙΟ,ΚΕΡΚΥΡΑ,ΚΩΣ,ΡΟΔΟΣ,ΚΑΒΑΛΑ,ΜΟΔΑ,ΔΡΑΜΑ,ΣΕΡΡΕΣ,ΕΥΡΥΤΑΝΙΑ,ΠΑΡΓΑ,ΚΕΦΑΛΟΝΙΑ, ΙΩΑΝΝΙΝΑ,ΛΕΥΚΑΔΑ,ΣΠΑΡΤΗ,ΠΑΞΟΙ
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MOSCHOPOULOS. RHABDAS 551<br />
<strong>of</strong> Smyrna, arithmetician and geometer, <strong>to</strong>v 'Pa/3oi).<br />
(a) On the left hand :<br />
for 1, half-close the 5th finger only;<br />
„ 2, „ „ 4th and 5th fingers only<br />
., 3, „ ,, 3rd, 4th and 5th fingers only<br />
„ 4, „ „ 3rd and 4th fingers only<br />
;<br />
„ 5, „ „ 3rd finger only<br />
„ 7, close the 5th finger only;<br />
„ 8, „ „ 4th and 5th fingers only<br />
„ 9, „ „ 3rd, 4th and 5th fingers only.<br />
(6) The same operations on the right hand give the thousands,<br />
<strong>from</strong> 1000 <strong>to</strong> 9000.<br />
(c) On the left hand :<br />
for 10,<br />
apply the tip <strong>of</strong> the forefinger <strong>to</strong> the first joint <strong>of</strong><br />
the thumb so that the resulting figure resembles o~ ;<br />
1<br />
A similar description occurs in the works <strong>of</strong> the Venerable Bede<br />
(' De compu<strong>to</strong> vel loquela digi<strong>to</strong>rum ', forming chapter i <strong>of</strong> De temporum<br />
ratione), where expressions are also quoted <strong>from</strong> St. Jerome (d. 420 A. d.)<br />
as showing that he <strong>to</strong>o was acquainted with the system (The Miscellaneous<br />
Works <strong>of</strong> the Venerable Bede, ed. J. A. Giles, vol. vi, 1843, pp. 141-3).<br />
552 COMMENTATORS AND BYZANTINES<br />
for 20, stretch out the forefinger straight and vertical,<br />
keep fingers 3, 4, 5 <strong>to</strong>gether but separate <strong>from</strong> it<br />
and inclined slightly <strong>to</strong> the palm ; in this position<br />
<strong>to</strong>uch the forefinger with the thumb<br />
„ 30, join the tips <strong>of</strong> the forefinger and thumb<br />
;<br />
„ 40, place the thumb on the knuckle <strong>of</strong> the forefinger<br />
behind, making a figure like the letter T ;<br />
„ 50, make a like figure with the thumb on the knuckle<br />
<strong>of</strong> the forefinger inside<br />
;<br />
„ 60, place the thumb inside the forefinger as for 50 and<br />
bring the forefinger down over the thumb, <strong>to</strong>uching<br />
the ball <strong>of</strong> it<br />
„ 70, rest the forefinger round the tip <strong>of</strong> the thumb,<br />
making a curve like a spiral<br />
„ 80, fingers 3, 4, 5 being held <strong>to</strong>gether and inclined<br />
at an angle <strong>to</strong> the palm, put the thumb across the<br />
palm <strong>to</strong> <strong>to</strong>uch the third phalanx <strong>of</strong> the middle<br />
finger (3) and in this position bend the forefinger<br />
above the first joint <strong>of</strong> the thumb<br />
„ 90, close the forefinger only as completely as possible.<br />
(d) The same operations on the right hand give the hundreds,<br />
<strong>from</strong> 100 <strong>to</strong> 900.<br />
The first letter also contains tables for addition and subtraction<br />
and for multiplication and division ;<br />
as these are said<br />
<strong>to</strong> be the ' invention <strong>of</strong> Palamedes ',<br />
we must suppose that<br />
such tables were in use <strong>from</strong> a remote antiquity.<br />
Lastly, the<br />
first letter contains a statement which, though applied <strong>to</strong><br />
particular numbers, expresses a theorem <strong>to</strong> the effect that<br />
(a + 10^ + ... + 10 m aJ (6 + 10^ + ... + \0 n b n )<br />
is not > 10 W+W+2 ,<br />
where a , a Y<br />
... 6 , b x<br />
... are any numbers <strong>from</strong> <strong>to</strong> 9.<br />
In the second letter <strong>of</strong> Rhabdas we find simple algebraical<br />
problems <strong>of</strong> the same sort as those <strong>of</strong> the Anthologia Graeca<br />
and the Papyrus <strong>of</strong> Akhmim. Thus there are five problems<br />
leading <strong>to</strong> equations <strong>of</strong> the type<br />
x x<br />
— + - + ...= a.<br />
n<br />
m