271 
the Stability of Ships. 
OD is an abscissa, being a portion of the axis of this curve, 
and the parameter is assumed = OD. If, therefore, an ordi- 
nate BD, or DA, is drawn through the point D perpendicu- 
lar to DO; the lines DB, DO, and DA, will all be equal. 
Let BD be divided into 10 equal parts, * considered in this in- 
stance as an abscissa, on which, at the points of division, the 
several ordinates are erected perpendicular to BD, denoted in 
the figure by the letters a , b, c, d , &c. If BA is assumed = 100 
equal parts, DB, DO, and DA, are each == 50, and the common 
interval between the ordinates = 5; the numerical values of 
the successive ordinates a , b> c, & c. are expressed in the an- 
nexed table. 
According to the Rule 1. making 
the sum of all the ordinates, or P 
= 466.134,1, the sum of the first and 
last ordinate, or S = 50 : r = 5 = the 
common distance of the ordinates. 
The area BDO = P — — x r 
2 
= 2205.670 
correct area = -~ x x 8 — 2222.222 
Difference f or error of 
the approximation = 16.552 
* Any line being assumed in a curve as an abscissa, lines drawn parallel to each 
other, and intercepted between the abscissa and the curve, are termed ordinates. To 
exemplify these rules for approximating to the areas of curvilinear spaces, it was ne- 
cessary to consider the ordinates as being drawn in some cases parallel, and in others 
perpendicular, to the axis. 
f It must not be concluded, from this instance, that the errors in measuring cur- 
vilinear areas by the Rule t. will be usually so great as 16 parts in 2222. In applying 
the Rule x. to this parabolic space, the quick variation of curvature in some parts of 
the curve, causes the space so measured to deviate more from the truth than would, 
happen in ordinary cases, such as commonly occur in practical subjects. If, instead of 
a 
= 50.0000 
b 
= 50.0000 
c 
= 49-99.99 
| d 
= 49-9967 
e 
= 49-9 6> 72 
f 
= 49.8047 
g 
= 49.1602 
h 
' = 47-1175 
i 
= 41.6113 
k 
= 28.4766 
l 
= 0.0000 
Sum of all the 
ordinates 
= 466.1341 
