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

116 HISTORY OF THE THEORY OF DETERMINANTS

SMITH, Ii. J. S. (1876).
[On the value of a certain arithmetical determinant. Proceed.
London Math. Soc., vii. pp. 208-212; or Collected Math. Papers,
ii. pp. 161-165.]
The determinant in question is that in which the element in the
(r, s)th place is the greatest common measure of r and s, namely,
1 1 1 1....
1 21 2....
1 1 3 1....
1 2 1 4....
and Smith's theorem is that the determinant of the A"t order, D,
say, is equal to
V(1).,2(2). V(3).... V(n),
where V/r(r) is the number of integers not greater than r and prime
to r. Recalling the fact that
+(n)= 12it-A1 ) ) *....
= n n n.- n
= n- ---.....+ + -- +.....
Pi P2 P1P2 P1P3
where the p's are the prime factors of n, he performs the operation
of subtracting from the nth row the rows whose order-numbers are
n n
P1) P2.
adding those whose order-numbers are
n n
PlP2' PiP3'
subtracting those whose order-numbers are
n
PlP2 ' P3. '
and so on: and this being done he is able to show that all the
elements of the nth row vanish except the last, which becomes Vr(n).