MENSURATION is the art of ascer taining the contents of superficial areas, or planes ; of solids, or substantial ob jects; and the lengths, breadths, &c. of various figures ; either collectively or abstractedsy. The mensuration of a plane superScies, orsurface,lyin g lev el between its several boundaries, is easy : when the figure is regular, such as a square, or a parallelogram, the height, multiplied by the breadth, will give the superficial con tents. Thus, if a table be 5 feet 2 inches in length, by 4 feet 1 inch in breadth, multiply 62, (the number of inches in 5 feet 2 inches) by 49, (the number of i inches in 4 feet 1 inch,) the result will she w the number of square inches; which, being divided by 144, (the number of square inches in a square foot,) will ex hibit the number of square feet on the surface of the table. Whatever balance may remain, may either be left as frac tienal, or hundred and fbrty.fourth parts; or, being divided by 36, may be made to show the numbers of quarters of square feet, beyond the integers produced by the first division.
For instance, multiply 62 inches by 49 inches 558 248 f Divide by 144)3038)21 ; 288 158 144 14 mom In regard to triangles, their bases mul tiplied by half their heights, or their heights by half their bases, will give the superficial measure. But it is necessary to caution our readers not to measure by the oblique line of a triangle, considering it as the altitude : a reference to the arti cle Glosses: will show, that the height of a triangle is taken by means of a per. pendicuLsr to the base, limited by a paral lel to the latter, which exactly includes the apex, or summit.
Any rectangular figure may have its sur face estimated, however numerous the sides may be, simply dividing it into triangles, by drawing lines from one angle to another, but taking care that no cross lines be made : thus, if a triangle should be equally subdivided, it may be done by one line, which must, however, be drawn from any one point to the centre of the opposite face. A four-sided figure will be divided into two triangles, by one ob lique line connecting the two opposite angles : a five-sided figure (or pentagon) by two lines, cutting as it were one tri angle out of the middle, and making one on each side : a six-sided figure (or hex agon) will require three diagonals, which will make four triangles : and so on to any extent, and however long, or short, the several sides may be respectively.
With respect to the form and proper. ties of various figures, we refer our read ers to the head of GEOXITHY, where all that relates thereto is pointed out, and the commutations they undergo, when their contents or areas are measured, will be distinctly seen.
The most essential figure is the circle, of which mathematicians conceive it im possible to ascertain the area with per fect precision, except by the aid of loga rithmic and algebraic demonstration. It may be sufficient in this place to State, that 814 of the diameter will give the side of a square, whose area will be cor respondent with that of a circle having 10 for its diameter. Therefore, as the diameter may be easily divided, either arithmetically, or mechanically, into ten equal parts, and one of those parts int° seventeen, by taking 8 integers, and 10 of the 17th portions, the aide of such a square may be easily demonstrated. Where a circle is small, its scale may be extended by an oblique line, which may be made to any extent, as shevrn in the fig. 7, Plate X. ?Aiwa where A B is the diameter of a circle, and A C the oblique line, lying betwesp the perpendiculars that would fall on A B. If A C be divid ed into- any number of parts, perpendi culars drawn from A B to the points of division, as a b, c d, will divide the diame ter exactly, in the same proportions' as AC is divided. The radius, or semidia meter of a circle, also gives us the means of forming a square corresponding with its area. Having drawn the whole diame ter, A B, fig. 8, take the radius, C B, and set it off from B to D ; from which mea sure another radius, at right angles with C B, to wherever it may fall, i e, at E, on the diameter : the hypothenuse, B E, will give the side of the square sought.