A history of Greek mathematics Vol.II from Aristarchus to Diophantus by Heath, Thomas Little, Sir, 1921

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 /Y> 1
l+-(®-l) = 2 + -{ X -(l+- + 2)},
which gives x = (n — l) 2 ;
therefore each of (n — 1) sons received
(n-l).
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
(n 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 discovered
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
550 COMMENTATORS AND BYZANTINES
instance of Nicolas Rhabdas, presently to be mentioned) a
treatise on magic squares ; he showed, that is, how the numbers
1 , 2, 3 . . . n 2 could be placed in the n 2 compartments of
a square, divided like a chess-board into n 2 small squares, in such
a way that the sum of the numbers in each horizontal and
each vertical row of compartments, as well as in the rows
forming the diagonals, is always the same, namely \n (n 2 + 1).
Moschopoulos gives rules of procedure for the cases in which
n = 2 m + 1 and n = 4 m respectively, and these only, in the
treatise as we have it ; he promises to give the case where
n = 4m+2 also, but does not seem to have done so, as the
two manuscripts used by Tannery have after the first two cases
the words reAo? rod avrov. The treatise was translated by
De la Hire, 1 edited by S. Gunther, 2 and finally edited in an
improved text with translation by Tannery. 3
The work of Moschopoulos was dedicated to Nicolas Artavasdus,
called Rhabdas, a person of some importance in the
history of Greek arithmetic. He edited, with some additions
of his own, the Manual of Planudes; this edition exists in
the Paris MS. 2428. But he is also the author of two letters
which have been edited by Tannery in the Greek text with
French translation. 4 The date of Rhabdas is roughly fixed
by means of a calculation of the date of Easter in the current
'
year ' contained in one of the letters, which shows that its
date was 1341. It is remarkable that each of the two letters
has a preface which (except for the words tt]v 8rj\a>cru/ ra>u kv
roh dpiOfioT? £r)Trjfj,dTcov and the name or title of the person
to whom it is addressed) copies word for word the first thirteen
lines of the preface
to Diophantus's Arithmetica, a piece
of plagiarism which, if it does not say much for the literary
resource of Rhabdas, may indicate that he had studied Diophantus.
The first of the two letters has the heading A con-
'
cise and most clear exposition of the science of calculation
written at Byzantium of Constantine, by Nicolas Artavasdus
1
Mem. de VAcad. Royale des Sciences, 1705.
Vermischte Untersuchungen zur Gesch. d. Math., Leipzig, 1876.
2
3
'Le traite de Manuel Moschopoulos sur les carres magiques' in
Annuaire de VAssociation pour* Vencouragement des etudes grecques, xx,
1886, pp. 88-118.
4
'Notices sur les deux lettres arithmetiques de Nicolas Rhabdas' in
Notices et extraits des manuscrits de la Bibliotheque Nationale, xxxii, pt. 1,
1886, pp. 121-252.