MENSURATION. Mensuration is that branch of applied mathematics which treats of the metrical relations of geometric figures, in particular of the length of lines, the magnitude of plane and solid angles, the area of surfaces and the volume of solids. The term is used both for the act and for the art of measuring geometric magnitudes. Mensuration is not usually treated as a separate branch of mathe matics, but occurs as an integral part of various subjects, such as plane geometry, solid geometry, trigonometry and integral calculus. For a dis cussion of the measurement of plane, diedral, spherical and solid angles and of the relations between the sides and angles of a triangle (plane or spherical) the reader is referred to the article on TRIGONOMETRY. The present arti cle will give formulas for lengths, areas and volumes of the simpler and the regular figures and methods of approximation for the more complex or irregular figures.
The measure of a geometric magnitude is its ratio to a fixed magnitude of the same kind selected as the unit of measurement. Through out this article the unit of area is assumed to be a square each side of which is equal to the unit of length, and the unit of volume is a cube whose edges are likewise of unit length. The purpose of a formula is to show how one of these numbers (ratios) may be found from certain others which are supposed to be known or obtainable. Thus formula (XIX) says that the ratio of the area of a circle to that of a square each side of which is a foot long (that is, the area of the circle measured in square feet) may be obtained by multiplying together the ratio of the circumference to the unit of length known as a foot, the ratio of the radius to the same unit of length, and the number TA. It is important to recognize that in using
any formula the units must all be of the same system. The number of acres in a field can not be found directly from formula (X) by multiplying together the number of rods in two adjacent sides: this would give the number of square rods in the field. Nor is the number of gallons in a barrel given by formula (LXV) when the radii are measured in feet or in inches, but the number of cubic feet or cubic inches, as the case may he. In order to change any measure from one system of units to an other the following table may be found useful: (For other multipliers see WEIGHTS AND MEAstutEs). The second column of figures serves to change the unit in the opposite way to that indicated by the rest of the table: thus to change meters to feet multiply the number of meters by 3.28087; for instance, 10 meters is equal to S2.8087 feet.
In many cases the computation of area or of volume may he accomplished most easily by mechanical means. For the measurement of the area of plane figures ingenious and effective instruments known as planimeters have been invented. Two historic instances are the find ing by Galileo of an approximate value for the area of a cycloid by cutting it out of a sheet of copper and weighing the model, and the dis covery by Archimedes of a fraud in Hierei new crown through measuring its volume }- submersing it in water and measuring the water displaced. To apply this last method, multiply the number of ounces of water by 1.73, or less accurately by I, and the result will be the num ber of cubic inches in the object; or, measure directly the volume of the water displaced.