1 14 Mr. Atwood’s Propositions determining the Positions 
The distance between the planes NXP, jixp, is the line 
Xx = Xn — Fp ; Xx produced, is the line in which the two 
sections of the water intersect each other, and is therefore 
coincident with the water’s surface, and is parallel to the 
longer axis. The dimensions of the vessel being supposed 
known, the lines AB, NI, will be known in fig. 2 : from these 
data the lines NX, PX, (fig. 2. and 28.) are to be assumed by 
estimation, and the angle NXP being given by the supposition, 
the area NXP is known from the rules of trigonometry, and 
the area PTNP may be inferred by the known methods of 
approximation.* 
In like manner the area xptn is to be determined, and a 
mean of the two areas being multiplied into the thickness or 
* Stirling. De Inter polatione Serierum, prop. xxxi. Chapman. Traite de la 
Construction des Vaisseaux, ch. i. 
Methods of approximating to the areas of curves, founded on the differential serieses, 
are given by several authors, particularly by Stirling and Simpson. Admiral 
Chapman proposes a very ingenious method of approximation, depending on the 
properties of the parabola ; either is sufficiently exact for the purposes of practical 
geometry, as appears by the instance inserted underneath: but of the two methods 
that of Mr. Stirling is the most correct. The two methods of approximation 
are severally applied and compared in the following example of finding the curvilinear 
area, which is comprehended between an arc of 30° and the radius, sine, and cosine 
of the said arc : to obtain this area by approximation, 5 equidistant ordinates are 
given ; i. e. 1st. ordinate 2= radius ~ 8, ad. — v' 63 : 3d. v' 60, 4th. rz ^55, 
5th. rz V' 48. 
The approximate area is. 
According to the method According to the method 
of Sti rling, - 30.61153 of Chapman, - 30.61131 
Correct area - - 30.61156 30.61156 
Error of approximation — .00003 — .00025 
The same method by which the areas of curves are found by approximation, may be 
applied with equal exactness to determine the solid contents of space, and the position 
•f the centre of gravity. 
