P. HISTORY OF ' AATHEMATICAL - School of Mathematics

86 A HISTORY OF MATHEMATICAL NOTATIONS
Arithmetic. He says: u •••• the continued product of l'2345678X
(1'01)4(1 '001)5(1'0001)8 which may be written 1 '2345678~0, 4, 5, 6, =
1'291907. The arrow ~ divides the coefficient 1 '2345678 and the
powers of 1'1,1'01,1'001,1'0001, &c; 0, immediately follows the arrow
because no power of 1'1 is employed; 6, is in the fourth place after ~,
and shows that this power operated upon periods of four figures each;
5 being in the third place after ~ shows by its position, that its influence
is over periods of three figures each; and 4 occupies the second
place after ~ ....."1 A few years later Byrne! wrote: "Since any
number may be represented in theformN=2"10m~ul,'U2,U3,U4,u5, etc.,
we may omit the bases 2 and 10 with as much advantage in perspicuity
as we omitted the bases 1'1,1'01,1'001, &c. and write the above expression
in the form N = m~ "UI,U2,U3, U4, etc." "Any number N may be written
as a continued product of the form io-x (1- '1)01(1- '01)02(1­
,001)0' (1-.0001)0', etc." or lO m ('9)01('99)02("999) 0'("9999) 0',etc." "In
analogy with the notation used in the descending branch of dual arithmetic,
this continued product may be written thus 'VI'V2'V3'V4'V5'V8~m
where any of the digits VI, V2, V3, etc. as well as m may be positive or
negative."!
"As in the ascending branch, the power of 1O,m, may be taken off
the arrow and digits placed to the right when m is a + whole number.
Thus 'VI' 'V2 'V3 'v4tt: t; t; etc. represents the continued product ('9)0'
('99)0 2('999)0'('9999)0'(9)1'(99)12(999)/0." Dual signs of addition :t and
subtraction --+ are introduced for ascending branches; the reversion
of the arrows gives the symbols for descending branches.
458. Chessboard problem.-In the study of different non-linear
arrangements of eight men on a chessboard, T. B. Sprague! lets each
numeral in 61528374 indicate a row, and the position of each numeral
(counted from left to right) indicates a column. He lets also i denote
"inversion" so that i(61528374) =47382516, r "reversion" (each
number subtracted from 9) so that r(61528374) =38471625, p "perversion"
(interchanging columns and rows so that the 6 in first column
becomes 1 in the sixth column, the 1 in the second column becomes 2
in the first column, the 5 in the third column becomes 3 in the fifth
column, etc., and then arranging in the order of the new columns) so
that p(61528374) =24683175. It is found that 1,"2=r=p2=1, ir=ri,
ip=pr, rp=pi, irp=rip=ipi, rpr=pir=pri.
1 Oliver Byrne, Dual Arithmetic (London, 1863), p. 9.
2 Oliver Byrne, op. cit., Part II (London, 1867), p. v.
3 O. Byrne, op. cit., Part II (1867), p. x.
4 T. B. Sprague, Proceedings Edinburgh Math. Soc., Vol. VIII (1890), p. 32.