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.

AREAS AND LIMITS OF SUMS

3I

1)-.X
f(a)/ +f(a +) + )A +-* +fb-2 + f(b -/)- ) / - (.))ax=- R, (a)
Jr=4
f(a +/ z)l +f(a +2 /z)l + ** +f(b-/i)/ +f(b)/= f()^,=L. (b)
al+ Ac
This notation means that x is increased by Ax from term to
term, from a to (b - x) in the first summation, and from
(a + Ax) to b in the second. This implies that (b - a) is an
exact multiple of Ax; but that follows since bAa = 1, a positive integer by definition. It is not necessary, however, that
this be so, nor indeed that the Ax's be all equal, for inequalities
of the kind considered can always be devised by having
Ax, < (b- a)< A-r,.
The question of incommensurability will not be more closely
touched upon here.
Hence from (a) and (b),
/f(x) -x A 5 f(x) x.
6- 1-A~ at+ -Ax
But f - X - [f(b) -f(a)]Ax vanishes with A-.
ct t+.dv tt
a-Al)b- Axr =
Therefore A = aL.o f(X)Av = a ox,
by reasoning the same as before.*
25. Let us apply the general reasoning to a concrete example. Take the cuive y -= 1 x2, say, and let it be required to
find the area between the ordinates where = 3 and x= 6.
Use coordinate paper, drawing in the curve very nicely with an
irregular draughting curve. The points are located most expeditiously by means of a table of squares. Adhering to our notation, suppose we divide the base, (b-a), into three parts; then
Y1=.9, Y2f= I.6, y= 2.5, y4= 3.6, Ax-rhz =, a = 3, b 6, n = 3.
y,.h = 5.o < A < yjh 7.7.
1 2

* This method of quadratures, as explained in the last three articles, might be
termed the modern method of exhaustions.