MENSURATION is the name given to a branch of the application of arithmetic to geometry, which shows how to find any dimension of a figure, or its arca, or surface, or solidity, &c., by means of the most simple measurements which the ease will admit of. We need hardly say that a complete treatise on this science would involve every branch of mathematical science. We shall in this article collect together the most important rules, the method of using which will be obvious to all who can employ tho trigonometrical tables. By the length of a line we mean the number of linear units contained in it, and by its square and cube the number of units multiplied by itself once and twice.
The measurement of lengths and directions resolves itself for the most part into the determination of a side or angle of a triangle, when other sides or angles are given. The triangle may be either on a plane or ou a sphere ; but we refer the latter to SPUERE, since the use of spherical trigonometry can only be well explained in connection with astronomy. Let a, b, c be the sides of a triangle, and A, n, o the opposite angles. If the triangle be right angled at o, we have tho following formuhe The area of a rectangle (in square units), and that of a parallelogram, is the product of the units in the base and perpendicular distance of the opposite sides. But if two sides only be parallel, half the sum of the parallel sides must be multiplied by the perpendicular distance between them. In other cases, the figure must be measured by dividing it into triangles, except when it is either a four-sided, figure capable of inscription in a circle, or a regular polygon. Every triangle is half of the rectangle contained by any one of its sides, and the perpendicular let fall from the opposite vertex.
If a, 5, c, and d be the sides of a four-sided figure inscribed in a circle, and a their half-sum, the area is Tables connected with this subject are given in the article REGULAR FIGURES. For the method of measuring irregular areas, see QueD RATTMES, METHOD OF.
The whole of the measurement of the circle depends upon the ratio of the circumference to the diameter, which is called r, and is 3.1415927 very nearly, or 27 roughly, or m very nearly. [ANGLE.]
So many simple derivations from this number are practically useful, that we shall give a table of them, accompanied by their logarithms, first giving a method of multiplying and dividing by a-, which is a correction of the use of V. To multiply by 71-, multiply by 22 and divide by 7; from the result take one-eighth of the hundredth part of the multiplicand as a correction; the result is too great only by about its 200,000th part. To divide by a-, multiply by 7, divide by 11 and 2, and to the result add the eighth part of the thousandth part of the dividend; the result is too small by very nearly its 100,000th part To find the circumference from the diameter, multiply by a.; to find the diameter from the circumference, multiply by 1 : w; to find the area from the diameter, multiply the square of the diameter by w : 4; to find the area from the radius, multiply the square of the radius by sr ; to find the diameter from the area, multiply the square root of the area by w); to find the area from the circumference, multiply the square of the circumference by 1 : 4 w; to find the circumference from the area, multiply twice the square root of the area by a/ ; to find the ordinate perpendicular to a diameter, take the square root of the product of the segments into which it divides the diameter.
To find the area contained between two concentric circles, multiply the product of the sum and difference of the radii by T.
The are of a circle and it subtended central angle are connected as follows : the arc which is equal to the radius subtends an angle of 57°1 very nearly ; or it may be easily remembered as 57 degrees and three tenths of a degree, diminished by one-fourth of a minute and ono-fifth of a second, being 57' 17' or 200264"1. To find an angle from Its arc (the radius being known), multiply the are by 571, and divide by the radius ; the result is too great by about three-quarters of its 10,000th part, and is in degrees and decimals of a degree. To find the arc from its angle, turn the angle into degrees and decimals, multiply by the radius, and divide by 571 : tho result is now too email by about three-quarters of its 10,000th part.