Mr. Atwood's 'Disquisition on 
! 266 
result multiplied into the common distance between the ordi- 
nates will be the exact area of the figure, considered as con- 
sisting of trapezia, and an approximate value of the curvilinear 
area in which the said trapezia are inscribed. This rule he pro- 
fesses not to be a very correct approximation, but such as may 
be deemed sufficient for most practical mensurations. It must 
be acknowledged, that in mensurations independent of others, 
the errors arising from this rule are often not considerable, 
(in many cases they are very small;) but, considering that in 
naval mensurations, areas obtained by approximation are ne- 
cessarily the data from which other results are to be inferred, 
also by approximation, a doubt may arise whether the errors 
thus accumulated may not, in some cases, become too great ; 
at least it may not be improper to be provided with rules which 
may be relied on, as approximating more nearly to the true 
measures of areas. 
Let the same area be measured by the rule opposite 3 ordi- 
nates, according to which it is supposed that the curve line 
ABC coincides with the arc of the conic parabola. By this 
rule, the area AA'C'CA = a~j-4fb~j-cx - : also the area 
3 
CC'E'EC = c + 4, d + e and the area EE'G'GE 
= e + 4 / + g * ; adding these three areas together, the 
sum is the area AA'G'GA= a + ^b-\-2c ^d-\-2e 4 /-f g 
r 
X ~. 
3 
This rule is the same with that which Mr. Simpson has de- 
monstrated in his Essays, page 109, from the properties of the 
conic parabola, perhaps not noticing that it was to be found in 
Mr. Stirling's table of areas. 
