ELEMENTS OF GEOMETRY AND MENSURATION. PART III. GEOMETRY COMBINED WITH ARITHMETIC. (MENSURATION.) THE application of Arithmetic to Geometry enables us to perform various calculations and measurements with respect to lines, lengths, distances, areas, angles, &c., and is commonly called MENSURATION, that is, the Art and Science of Measuring. GENERAL PRINCIPLES OF MEASUREMENT. I. OF LINES. 214. The Arithmetical Measure of a line or length is the ratio which that line, or length, bears to another line or length taken for the unit or standard of measure. Thus, if a certain line, or length, be called a foot, then another line, or length, which contains the former exactly three times, that is, which is just three times as long, will be 3 feet. And if 1 represent the former, 3 will represent the latter, line. Also another line which contains the same unit five-and-a-half times, will be represented by 5; and so on. 215. It is not necessary that the unit of lineal measurement be the line, or length, which we call a foot; it may be any other line, or length, taken at pleasure. But it is necessary, when numbers are used to represent Geometrical magnitude, in every case to know, and to bear in mind, what unit, or standard, has been employed. PART III. ж 1 Thus, if a certain line, or length, be represented by 13, we know nothing about it until we know what the unit is. If the unit be an inch, then the given line is 13 inches; or, if the unit be a mile, then the given line, or length, is 13 miles: and so on. 216. Hence, it is obvious, that much practical benefit arises from using those units, or standards, only which are well known, as an inch, a foot, a yard, a mile, &c. For thus we are enabled to communicate to others, by means of a few words and symbols, a true notion of the magnitude of any line, length, or distance, with which we are concerned, when it is once known to ourselves. Thus, if we wish to inform a friend that we have walked a long distance within a certain time, by stating that we walked 100 miles in 3 days, we give him an accurate notion of our achievement, because a mile and a day are units, the one of length, and the other of time, with which he is supposed to be well acquainted. 217. It might, indeed, without the use of either Arithmetic or Geometry, be said that it is a long distance from London to Edinburgh. Edinburgh. But this, in fact, expresses nothing which has any significance, because another person might as truly assert, that the same distance is short. Both may be right, having taken different units of measurement. For the distance between London and Edinburgh is great compared with the length of a man's foot, that is, when the unit is a foot; but the same distance is small when compared with the circumference of the Earth, or the distance to the Moon. And so, then, let it be borne in mind, that the measuring of a line, or length, is simply the comparing it with some other line, or length, taken as a standard; and any proposed line, or length, is great or small only in respect of some other line, or length, with which the former is compared. 218. To measure a given straight line or length. (1) When the given straight line is accessible in every part of it, let it be represented by AB, where A and B denote the extreme 4 points. Take a foot-rule or yard-wand, B or some other convenient standard of measure, and lay it along AB, so as to have one of its ends coinciding with A. Mark where the other end meets the line, or length; from that point repeat the operation, as was done from A; and so on, until the standard has been laid along the whole line from A to B. This process enables us to see and count how many times the standard, or unit of measure, is contained in the given line; and that number of times is the measure of the line; for it is the ratio which the given line bears to the unit of measure. This number may be either whole or fractional, according to circumstances. In some cases the unit will be contained an exact integral number of times in the given line; in others so many times, and parts of a time. And, in order that the fractional part may be readily determined, the standard, or unit, is divided into a certain number of equal parts, (as the foot-rule into 12 equal parts, called inches,) and each of these parts again into a certain number of equal parts; and so on, to any required degree of minute subdivisions. So that, if, for example, AB contain the foot-rule 3 times and five-twelfths of another time, then the measure of AB is 3 feet, or 3 feet 5 inches. All this is so obvious to the senses in practice, that it requires no further illustration. (2) For short lines the draughtsman more commonly makes use of the compasses, opening the compasses so as to separate their feet to the exact distance AB, he then applies that distance to the face of his flat ruler, which is divided into inches and parts of an inch; and thus he measures the line AB by noting how many inches and parts of an inch, (or, if AB be less than an inch, how many parts of an inch,) the compasses embrace. Or, he places the edge of the graduated ruler along the line itself, which is to be measured, and observes at once with how many divisions of the ruler the given line coincides, and so measures it. This is often the most expeditious method. (3) Lastly, for long lines a tape is commonly used, which is divided into feet and inches, and wound on a reel. One end of the tape is held at one end of the line, or length, to be measured, and the tape is then unwound, until, being tightly stretched, there is sufficient of it to cover the line in its whole extent. The figures marked on the tape, where it coincides with the other end of the line, express the length of the line in feet and inches. Or, if the length to be measured be greater than the whole length of the tape, it is only necessary to repeat the operation by successive measurements, as in the first case. The method of measuring still longer lines, or lengths, on the Earth's surface, as adopted by surveyors, will be given hereafter. 219. To measure a given crooked line or length. (1) If the crooked line consist of two or more straight lines joined together, it is obvious that the whole line may be measured by adding together the measures of the several lines taken A separately (determin ed as in the last Art.), which make up the whole. B C Thus, it is evident that the measure of the crooked line ABCD will be found by adding together the measures of AB, BC, and CD. Or, with the tape, it may often be done in one single measurement. For, if ABCD be the boundary of a rigid body, or if pegs be fixed at B and C, the tape may be tightly stretched so as to coincide with AB, BC, and CD, and thus shew at once the measure of ABCD. (2) If the line, or length, to be measured be a curved line, its measure may be found by carefully laying a string upon it throughout its whole extent, and then applying the foot-rule, or other standard, to find the length of the string stretched out into a straight line. Or, by means of a tape, a curved line, or length, may sometimes be measured at one step, since the tape combines in itself both the flexible string and the graduated measure. Thus, the woodman finds the girth of a tree, or the tailor the circumference of a man's body, in a moment of time. II. OF SUPERFICIES, SURFACES, OR AREAS. 220. In the same manner as the Arithmetical Measure of a line is the ratio which that line bears to another line, taken as the unit, or standard-so the Arithmetical Measure of a superficies, surface, or area (all which mean the same thing), is the ratio which that surface bears to another surface, taken as the unit, or standard, of superficial measure. And as the lineal foot was stated to be often the most convenient unit of length for measuring lines, so the square foot (that is, the square* of which each side is a lineal foot) is a common and convenient unit, or standard, of superficial measurement. Hence, taking this unit, the measure of a surface, or area, is the number of times which that surface, or area, contains a square foot; and that number will be sometimes a whole number, and sometimes fractional. For example, suppose the annexed diagram, ABCD, to be a miniature representation of a rectangle, of which the side AB is 4 feet, and the side AD is 1 foot; then, dividing AB into four equal parts (168, Part 11.) in E, F, G, and drawing Ee, Ff, Gg, parallel to AD, or BC, it is obvious that we have divided the rectangle into 4 equal squares, each of which is a square foot; therefore the rectangle ABCD is plainly equal to 4 square feet, that is, the measure of the rectangle is 4, when the unit is a square foot. 221. But as it often happens, that a given superficies, surface, or area, which it is proposed to measure, does not contain an exact integral number of square * It must be borne in mind that a square is not four straight lines of equal length and at right angles to each other, but the plane area included within those lines. |