272 Mr. Atwood’s Disquisition on 
If the same area is measured by the Rule 11. the process will 
be as underneath : 
Sum of all the ordinates - —466.1341 
Sum of the first and last, or S — 50.0000 
Sum of the 2d, 4th, 6th, &c. or P = 225.3355 
S + P = 2 75.3955 a 75- 3955 
Sum of the 3d, 5th, 7th, &c. (except the last) or Q — 190.7386 
and r being = 5, the area EDO = S-^P-f 2O x — = 222 1 .765 
Correct area - =2222.222 
Difference or error of the approximation = .457 
Let the same area be measured by the Rule m. the area 
DOKI, between the two ordinates a and b , being = 5 x 50 
— - 250, the remaining area, from the ordinate b to the ordinate 
/ = o, will be obtained from the following computation : 
The suni of all the ordinates = 416.1341 
Sum of the first and last, reckoning 
b the first, and the last / = o or S = 50.0000 
Sum of the 4th, 7th, 10th, &c, ( b being 
the 1st.) or P - - =97.0847 147.0847 
Sum of the 2d, 3d, 5th, 6th, &c. from 5, or Q = 269.0494 
the total area — DBO, (fig. 25.) a portion of it, which is contained between the ordi- 
nates a and g, should be measured, the area, computed from either of the three rules* 
would deviate very little from the truth, as appears from the following results. 
Areas contained between the ordinates a and g. 
computed by 
Rule 1. 
Rule 11. 
Rule hi. 
Areas 
H9 6 -74 
1497.17 
1497.13 
Correct area 
Difference or error of 
approximation 
1497. 20 
1497.20 
1497.20 
.46 
.03 
.07 
If the rule in the table of areas opposite 7 ordinates should be applied to measure 
the area between the ordinates a and^-, (fig. 25.) the result would be geometrically 
correct. 
