RULE

FOR PROCEEDING AFTER THE DIMENSIONS ARE TAKEN.

To the sum of the first and last ordinates add four times the sum of all the even ordinates, with double the sum of all the odd ones, except the first and last. Then multiply the total by one third part the equi-distance assigned, and the product will be the area between the extreme ordinates.

Lastly, for the area of the segments at the ends, multiply the sum of the first and last ordinates, by one third part the sum of the altitudes of the two segments, and the product will be the area of both, nearly, which being added to the former area, will give the superficial content of the whole figure.

THE AREA OF THE ABOVE CURVILINEAL FIGURE DETER

To find the Area of an Irregular plane Figure. Let ABCDEFG be a figure so irregular that its area cannot be found by any of the foregoing Rules.

If two straight lines not parallel be drawn in the curvature DG, and perpendiculars bisecting them be raised, those perpendiculars will meet in a point without the figure, which is the centre of the arc DG. Wherefore, if a chord be drawn from D to G, that part of the given figure is a segment of a circle.

In like manner it will be found that ADH, fkg, and CHE are sectors.

Also EHKF, and ABCH are quadrilaterals; and DHK is a triangle.

Evidently then, the given figure has been divided into figures that can be measured by the preceding problems, and the sum of the areas of those figures will be the area of the given figure.

In like manner any plane space whatever may be divided into portions, each of which shall fall under some one of the shapes treated of, in the preceding problems. No general Rule, however, can be given for the boundless variety of irregular figures that will occur in practice, and consequently much must be left to the discretion of the practitioner, which will always be great in proportion to his acquaintance with the manner of constructing the more common geometrical figures, and the correctness of his theory on measuring them.

The main object to be kept in view is to attain a MAXIMUM of accuracy with a MINIMUM of labour, and hence no more lines than are absolutely necessary, ought ever to be drawn.

SECTION II.

OF AREAS AN INCH IN DEPTH.

It is evident by the First Section that the number of square inches in the area of the plane figure constituting the base, or surface, will be the number of cubic inches in the solid space between the base and surface; hence further Rules on the subject would be

needless, were it not that those cubic inches are to be converted into exciseable measures, and that this may sometimes be done without finding the number of square inches in the area of the surface or base. Besides, as this species of Mensuration is one of the most useful parts of Gaging, since all areas in every figure whatever are considered as an inch in depth, and not, like in other branches of the Mathematics, as merely superficial, we shall, with this clearly understood, enter more fully into the spirit of the work before us.

Exciseable liquors are measured either by the Table of WINE MEASURE, or by the Table of ALE AND BEER MEASURE, the gallon being in both instances the lowest integer. For which reason, as a preliminary step, we shall subjoin these Tables, specifying the number of cubic inches in the several denominations, with the times any one lower denomination has to be taken to make a unit of a higher.

TABLE OF WINE MEASURE.

By Wine measure all sorts of wines and spirits, mead, cider, perry, verjuice, oil, and other liquids are bought and sold.

The fourth part of a gallon is called a quart, and contains 57.75 cubic inches.

The half of a quart is termed a pint, and is equal to 28.875 cubic inches.

Exciseable goods not falling under the head of liquors are generally reducible to some one of the following weights or measures.