s68 
Mr. Atwood’s Disquisition on 
only when the number of given equidistant ordinates is odd, 
to obtain the area when the number of ordinates is even, an- 
other rule, to be employed either singly, or in conjunction with 
the former, may be selected from Mr. Stirling’s table of areas. 
It is that which stands opposite 4, ordinates. 
Let it be proposed to measure the area AA'G'GA. Accord- 
ing to this rule, the area AA' D'DA = a -j- 36 -j- 3 c -| -d x 2T 
also the area DD'G'G A = d -j- 3 e -J- 3/ -f- g x ^ 
these two areas being added together, the sum will be the area 
AA'G'GA == a -f 3 b -j- 3 c + 2 d + 3 e + S.J + £ x T 
When the number of given equidistant ordinates is small, these 
theorems will be most conveniently used in the forms here given; 
but, when the ordinates are numerous, the trouble of arithme- 
tical computation will be considerably abridged, by employing 
them according to the general rules inserted underneath. 
These theorems for approximating to the values of areas, may 
be applied, with advantage, to the integration of fluxional quan- 
tities, the fluents of which cannot be obtained by direct methods; 
or, if obtained, requiring, very long and troublesome calculations.* 
Suppose z to represent the abscissa, of a curve, on which the. or- 
* On this principle, the rules of approximation here given are applicable to deter- 
mine the positions of the centres of gravity, both of areas and solid spaces. It y is put 
to represent the ordinate erected perpendicular to an absciss*, at the distance z from the 
initial point thereof, the fluent of yzi. (fig. 24.) will be the sum of the products ari- 
sing from multiplying each ordinate into the small increment z, and also into the dis- 
tance z from the initial point. And, since the area intercepted between the ordinates 
AA'and y is the fluent of yz , it follows, that the distance of the centre of gravity 
«f this curvilinear space from the ordinate AA', measured on the abscissa AT', is 
„ The approximate values of these fluents are obtained from the Rules i« 1 1. 
fluent yz 
