PLANUDES. MOSCHOPOULOS
549
the scholia to Eucl., Book X, the same method is applied.
Examples have been given above (vol. i, p. 63). The supposed
new method was therefore not only already known to the
scholiast, but goes back, in all probability, to Hipparchus.
Two problems.
Two problems given at the end of the Manual of Planudes
are worth mention. The first is stated thus: ‘ A certain man
finding himself at the point of death had his desk or safe
brought to him and divided his money among his sons with
the following words, “ I wish to divide my money equally
between my sons : the first shall have one piece and -|th of the
rest, the second 2 and ^th of the remainder, the third 3 and
|th of the remainder.” At this point the father died without
getting to the end either of his money or the enumeration of
his sons. I wish to know how many sons he had and how
much money.’ The solution is given as (n— l) 2 for the number
of coins to be divided and (n— 1) for the number of his sons;
or rather this is how it might be stated, for Planudes takes
n = 7 arbitrarily. Comparing the shares of the first two we
must clearly have
1 1 x 1
1 + -{x- 1) = 2 + -{»—(! + + 2)},
n ' n ' n
which gives x = {n— l) 2 ; therefore each of (n— 1) sons received
fa-1)-
The other problem is one which we have already met with,
that of finding two rectangles of equal perimeter such that
the area of one of them is a given multiple of the area of
the other. If n is the given multiple, the rectangles are
Ci 2 —1, n 3 ~n 2 ) and (n~ 1, n 3 — n) respectively. Planudes
states the solution correctly, but how he obtained it is not clear.
We find also in the Manual of Planudes the ‘proof by nine’
(i.e. by casting out nines), with a statement that it was dis
covered by the Indians and transmitted to us through the
Arabs.
Manuel Moschopoulos, a pupil and friend of Maximus
Planudes, lived apparently under the Emperor Andronicus II
(1282-1328) and perhaps under his predecessor Michael VIII
(1261-82) also. A man of wide learning, he wrote (at the
