2 6g 
the Stability of Ships. 
scissas being expressed by the equation y — a -\- px -\-qx 7 -\-tx\ 
&c. -f- ux n 3 (in which case, the ordinates are drawn parallel to 
the axis of the curve,) a measure of the area contained between 
the extremes of n + i ordinates will be obtained with geome- 
trical exactness, by computing from the rule in the table which 
is opposite the number of ordinates w + 1, supposing the table 
to extend to that number : but if, as it usually happens in cases 
which practically occur, that the nature of the curve is un- 
known, or the conditions in other respects different from those 
which are required for the mensuration of the area with perfect 
correctness, it becomes a question, which particular rule in the 
table should be adopted for inferring an approximate value of 
the area, since an exact quadrature is not obtainable. For this 
purpose, there are several reasons for preferring the rules oppo- 
site the number of ordinates 2, 3, and 4 to the others, which 
require a greater number of ordinates ; the common distance 
between them being the same. In the first place, the rules 
here pointed out are far less troublesome in the application ; a 
circumstance which ought to have weight, although of less im- 
portance than another consideration, which is, that the results 
derived from these rules, particularly from the two latter, will 
in general approximate as nearly to the true value, sometimes 
more nearly, than those which are obtained by calculating 
from the other more complicated theorems, unless the curve 
should happen to be such as admits of being correctly measured 
by any of the rules requiring a greater number of ordinates ; 
a circumstance not likely to occur in practical mensurations. 
Let it be proposed to measure by approximation any curvi- 
linear space AA'I'IA (fig. 24.) For brevity, let the successive 
ordinates A A', BB', CC', &c. be denoted by the letters a , b 3 c r 
