552 COMMENTATORS AND BYZANTINES 
for 20, stretch out the forefinger straight and vertical, 
keep fingers 3, 4, 5 together but separate from it i 
and inclined slightly to the palm; in this position 
touch the forefinger with the thumb; 
„ 30, join the tips of the forefinger and thumb ; 
„ 40, place the thumb on the knuckle of the forefinger 
behind, making a figure like the letter f ; 
„ 50, make a like figure with the thumb on the knuckle 
of the forefinger inside; 
„ 60, place the thumb inside the forefinger as for 50 and 
bring the forefinger down over the thumb, touch 
ing the ball of it; 
„ 70, rest the forefinger round the tip of the thumb, 
making a curve like a spiral; 
„ 80, fingers 3, 4, 6 being held together and inclined 
at an angle to the palm, put the thumb across the 
palm to touch the third phalanx of the middle 
finger (3) and in this position bend the forefinger 
above the first joint of the thumb; 
„ 90, close the forefinger only as completely as possible. 
{d) The same operations on the rigid hand give the hun 
dreds, from 100 to 900. 
The first letter also contains tables for addition and sub 
traction and for multiplication and division; as these are said 
to be the ‘invention of Palamedes’, we must suppose that 
such tables were in use from a remote antiquity. Lastly, the 
first letter contains a statement which, though applied to 
particular numbers, expresses a theorem to the effect that 
{% + 10a x + ... + 10 m a m ) {b 0 + 10+ ... + 10 n b n ) 
is not > 10 m+w+2 , 
where a 0 , a x ... b 0 , b 1 ... are any numbers from 0 to 9. 
In the second letter of Rhabdas we find simple algebraical 
problems of the same sort as those of the Anthologia Graeca 
and the Papyrus of Akhmim. Thus there are five problems 
leading to equations of the type 
x x 
—■ +—+... — a. 
m n