An introduction to the summation of differences of a function; an elementary exposition of the nature of the algebraic processes replaced by the abbreviations of the infinitesimal calculus, by B. F. Groat.

INVERSIONS AND DETERMINANTS
CHAPTER I
INVERSIONS OF ORDER
I. If several consecutive symbols of a sequence, as letters
of the alphabet, or numerals, be written in a line in any order,
then an inversion of natural order, or simply an inversion, occurs whenever any symbol follows another which it should
naturally precede. In. bac, 5647, 4321, JKIHG, 564213, there
are respectively I, 2, 6, 9, 12 inversions.
2. In considering any such line of symbols we shall number
their positions to the right beginning at the left. Then the
order of position of any symbol in the line is of the Ist, 2nd, 3d,
degree according as it occupies the ist, 2nd, 3d, position; and
so on. The order of a symbol, and the degree of the order,
is taken with reference to the natural sequence of the symbols in the line: it is the same as the order of the position
it would occupy if the symbols were arranged in natural order
in the positions of the line. It will be frequently convenient
to refer to a symbol, or position, as being even or odd, meaning
tihat the order of the symbol, or position, is of even or odd
(ld gree.
3. Theorem I. In any line of symbols capable oj arrangement
in natural sequence, if any two symbols be interchanZged tlze numzber
of ilversions in the line will be increased or decreased by an. odd
Tinmber.
The interchange of any two symbols occupying consecutive