The theory of determinants in the historical order of development, by Sir Thomas Muir.

490 HISTORY OF THE THEORY OF DETERMINANTS

HUGEL, T. (1876): STUDNICRA, F. J. (1876).
IDas Problem der magischen Systeme. 48 pp. Neustadt.]
'[Ueber das Verhiiltniss der magischen Quadrate zur DeterminantenTheorie. Sitzungsh.... Ges. d. Wiss. (Prag), pp. 269-271.]
The main fact noted by Studnic'ka is the vanishing of all determi-nants of the order 1 6n2, whose elements are identical with those of
magic squares constructed in accordance with the so-called second
-rule of Moschopoulos. For example, the operations
col4-Col1, col3-Col2,show that

a+1 /3-1
/3-4 a+6
a+8 /3-6
I/3-3a3

/3-2 a+4
a-17 /3-7 = 0,
/3-5 a+5
a+2 /3

where the elements of the determinant are those of a magic square
with the common sum 2 (a +/3+l1).
As for Hugel it suffices to note that in the course of his investigations he makes effective enough use of determinants, so-called and
actual.
MUIR T. (1877).
[Note on determinant 'expressions for the sum of a harmonical
progression. Proceed. R. Soc. Edinburgh, ix. pp. 361-363.]
The expression obtained for
1+ 1 1 1-
a a-Hb a+2b a+(n-1)b
is

1 - b - b - b
1 na -2b -3b
1 (n-1I)a a -3b
1 (n- 2)a a a
1 (n- 3)a a a
1 2a a a

- b.... - b
- 4b.... - (n-1i)b
-4b.... - (n- 1)b
-4b.... - (n-1i)b
a.... - (n-1I) b

+ a(a+b) -... (a+n-lb).

a.... a