26l 
the Stability of Ships. 
number of points situated in the same plane, he could ascertain 
the equation to the curve which would pass through them all : 
and, by means of this equation, was enabled to express the 
ordinate in the curve, corresponding to an abscissa of any 
given length, as well as the area intercepted between any two 
of the ordinates. This discovery the author himself considered 
amongst his happiest inventions. Amongst the various uses of 
this theorem, that of determining by approximation the areas 
of curvilinear spaces is not the least considerable : for, by 
this means, the fluents of fluxional quantities, not discoverable 
by any known rules of direct investigation, are found, to a de- 
gree of exactness fully sufficient for any practical purpose, and 
with very little trouble of computation. 
Mr. Stirling, in his treatise intitl ed' Methodus differentialis , 
has inserted a table for measuring curvilinear spaces termi- 
nated by parabolic curves, from having given 3, 5, 7, or 9 equi- 
distant ordinates, and the abscissae on which they are erected. 
The measures of the areas thus obtained are, under certain 
conditions hereafter stated, not approximations, but geometri- 
cally and strictly correct : the approximate values of curvili- 
near spaces, in general, are obtained from finding the correct 
areas terminated by parabolic lines which nearly coincide with 
the said curves, by passing through the extremities of the same 
ordinates. 
The subjoined table contains Mr. Stirling's rules for ex- 
pressing the areas of curvilinear spaces, from the conditions 
which have been mentioned ; also additional rules for measuring 
the areas which are included between the extremes of 2, 4, 6, 
or 8 equidistant ordinates : the whole of this table has been re- 
computed and verified. 
