ao 4 'M.r. Atwood's Disquisition on 
&c. respectively ; also, let the common distance between the or- 
dinates (fig. 23.) or AB' = B'C' — C'D' be = r : according 
to the theorem for measuring the area contained between two 
ordinates, or ~ x R, the curve line AB is supposed to coincide 
with the right line AB which joins the extremities of it; 
the space measured by this rule is the trapezium AA'B'BA; 
and, since A = a -f- b, and R = r, the area of the trapezium, or 
~xR = a^ftx-. 
According to the rule opposite 3 ordinates, the curve line 
ABC is supposed to coincide with a portion of the conic para- 
bola, the axis of which is parallel to the ordinates. And since, 
by this rule, the area == A -f- 4B x in which expression 
A = a -j- c, B = b, and R = 2 r; by substituting these values, 
the area AA' C' C A — If the curve ABC 
should actually be a portion of the conic parabola, the given 
ordinates being parallel to the axis of the curve, the area 
AA'C'CA will be measured by this rule with exactness abso- 
lutely perfect ; and, the more nearly the curve which terminates 
the area approaches to the form of the conic parabola, the more 
nearly will the result of calculating by this rule approximate to 
the true value of the area. But it is evident, that since in this 
approximation, the arc of a conic parabola is drawn through the 
points A, B, C, being the same points which terminate the or- 
dinates of the given curve, the difference between the parabolic 
area and that which is given must, in most (except extreme) 
cases, be next to an insensible quantity, when applied to prac- 
tical mensuration. 
In the mensuration of areas by the rule opposite 4 ordinates. 
