A history of Greek mathematics - Wilbourhall.org

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).
which
Moschopoulos gives rules of procedure for the cases in
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.
2
Vermischte Untersuchungen zur Gesch. d. Math., Leipzig, 1876.
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.