550 COMMENTATORS AND BYZANTINES
instance of Nicolas Rhabdas, presently to be mentioned) a
treatise on magic squares ; he showed, that is, how the num
bers 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 \u {n 2 + 1).
Moschopoulos gives rules of procedure for the cases in which
u = 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 = 4 m + 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 réAoç tov avrov. The treatise was translated by
De la Hire, 1 edited by S. Günther, 2 and finally edited in an
improved text with translation by Tannery. 3
The work of Moschopoulos was dedicated to Nicolas Arta-
vasdus, 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 rrjv drjXœcnu ruin' kv
rois àpLÔ/xoîs ^qrqgdroiv and the name or title of the person
to whom it is addressed) copies word for word the first thir
teen 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 Dio-
phantus. 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 Mém. de VAcad. Royale des Sciences, 1705.
2 Vermischte Untersuchungen zur Gesch. d. Math., Leipzig, 1876.
3 ‘ Le traité de Manuel Moschopoulos sur les carrés magiques ’ in
Annuaire de VAssociation pour Vencouragement des études grecques, xx,
1886, pp. 88-118.
4 ‘Notices sur les deux lettres arithmétiques de Nicolas Rhabdas’ in
Notices et extraits des manuscrits de la Bibliothèque Nationale, xxxii, pt. 1,
1886, pp. 121-252.
