MENSITRA'TION, the name of that branch of the application of arithmetic to geometry which teaches, from the actual measurement of certain lines of a figure, how to find by -calculation, the length of other lines, the area of surfaces, and the volume of solids. The -determination of lines is, however, generally treated of under trigonometry (q.v.), and surfaces and solids are now understood to form the sole subjects of naensuration. As -the length of a Iine is expressed by comparing it with some well-known unit of length, such as a yard, a foot, an inch, and saying how many such units it contains, so the extent of a surface is expressed by saying how often it contains a corresponding super ficial unit, that is, a square whose side is a yard, a foot, an inch; and the contents of solid bodies are similarly expressed in cubes or rectangular solids having- their length, breadth, and depth, a yard, a foot, an inch. To find the leng-th of a line (except in eases where the length may be calculated from other known lines, as in trigonometry) we have to apply the unit (in the shape of a foot-rule, a yard measure, a chain), and dis eover by actual trial how rnany units it contains. But in measuring a surface or a solid -we do not require to apply an'actual square board, or a cubic block, or even to divide it into such squares or blocks; we have only to measure certain of its boundary-lines or dimensions; and from them we can calculate or infer the contents. To illustrate how this is done, suppose that it is required to determine the area of a rectangular figure ABCD, of which the side AB is 7 in., and the side AC 3 inches. If AC be divided at the points F and E into 3 portions, each 1 inch long, and parallels be drawn from F and E to AB or CD; and if AB be similarly- divided into 7 parts, of 1 inch each, and par allels be drawn to AC or BD .through the points of section, then the figure will be
divided into a number of equal squares or rectangular figures, whose length and breadth are, each 1 inch; and as there are 3 rows of squares, and 7 squares in each row, there must be in all 7X3, or 21 squares. In general tern-is, if a and b be the lengths of two adjacent sides, there are a rows of little squares, and b squares in each row. Hence Om area of a rectangle = the produet of two adjacent sides.
The areas of other figures are found from this, by the aid of certain relations or prop erties of those figures demonstrated by pure geometry; for instance, the area of a parallelo gram is the same as the area of a rectangle having the same base and altitude, and is there fore equal to the base multiplied by the, height. As a triangle is half of a parallelog-ram, ;the rule for its area can be at once deduced. Irregular quathilaterals and polygons are measured by dividing them into triangles, the area of each of which is separately calcu lated. For the area of the circle, seatatctE. By reasoning similar to what has been employed in the case of areas, it is shown that the volume of a rectannular parallelo piped or prism is found in cubic inches by multiplying together the lengai, breadth, and depth in inches; and the oblique parallelopiped, prism, or cylinder, by multiplying the area of the base by the height.