A history of Greek mathematics - Wilbourhall.org

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552 COMMENTATORS AND BYZANTINES
for 20, stretch out the forefinger straight and vertical,
keep fingers 3, 4, 5 together but separate from it
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 T ;
„ 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, touching
the ball of it
„ 70, rest the forefinger round the tip of the thumb,
making a curve like a spiral
„ 80, fingers 3, 4, 5 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 right hand give the hundreds,
from 100 to 900.
The first letter also contains tables for addition and subtraction
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
(a + 10^ + ... + 10 m aJ (6 + 10^ + ... + \0 n b n )
is not > 10 W+W+2 ,
where a , a Y
... 6 , b x
... are any numbers from 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