To lay out a given area in a rectangular form, having the length to the breadth in a given ratio THE box of instruments should contain a pair of dividers; compasses, with lengthening bar, and pen and pencil; a bow-pencil and bow-pen; an ivory protractor, as shewn in the annexed drawing; a twelve-inch ivory or box scale, with a two-inch offset scale to match. The protractor is used for angular measurement-thus, suppose it were required to take off an angle of 60° (see Geometrical Problem No. 3, p. 7), at the point A, place the protractor, so that its centre A should coincide with, or fall upon A, and the line AB upon the line AB. It will be observed, that the degrees are numbered both ways round the protractor. An angle may therefore be taken either from left to right, or from right to left, as the case may require; but as it is divided only to degrees, the minutes must be guessed at. Now to lay off the given angle of 60 degrees from the line AB, look for 60°, counting from B B right to left, and make a mark on the paper at C. A line drawn from A through C will make with the line AB the required angle of 60°, which is termed the angle BAC or CAB, the centre letter always signifying the angular point. Any other angle may be taken in the same way-thus, the angle BAE would be an angle of 90 degrees, called a quadrant or right angle. There is a larger protractor furnished with a vernier, but it is an expensive instrument, and is not necessary, except where great accuracy is requisite. The inner portion of the protractor called the diagonal scale of equal parts is used for measuring distances. It is arranged for two scales, the one (called the scale of forty) being either 4, 40, 400 chains, yards, or feet, to the inch; the other (called the scale of twenty) being either 2, 20, or 200 chains, and each of these parts is subdivided into ten. Now, if the larger divisions be called chains, each of the small divisions, as seen at either end of the scale, will be ten links. The whole of the upper line of the diagonal scale will read six chains, or 5.90 chs., 5·80 chs., &c., according as you contract the compasses by one or more of these ten divisions on the right. The lower line will read 12 chs., or 11.90, 11.80, &c., as you contract them on the left. For anything less than 10 links the other intermediate horizontal lines must be used; the distances upon them being determined by the intersection of the diagonal lines at either end. Now looking at the upper line (scale of twenty) where the large divisions number from right to left, it will be seen clearly that, if from 1 to a be one chain, one yard, one foot, &c., then from a to in the smaller division 1 is 10 or of one chain or yard-from a to is 20, or 6 ; from a to is 60 or; and that from 1 of the larger divisions to of the smaller is therefore 1.20; from 1 to 6 is 1.60; and so on. To mark off, therefore, any distance given you in units and tenths, extend the compasses from the larger division corresponding to the given unit, to that of the smaller division corresponding to the tenths, whether these units be chains, yards, &c. This distance measured on the top line will be the distance required. Again, by following any diagonal line downwards (scale of 2) it will be seen at the bottom to become more to the right than it was at the top-that is, in the whole breadth of the scale 10 links or hundredths of the unit are gained; now as this breadth is divided into 10 equal parts by intermediate horizontal lines, at one line down, it will be '01 or Toʊ, or (one link, if the unit is a chain); at two lines down, 02 or To (two links), and so on; the numbers of the horizontal lines are marked at each end. Hence, for the chains (scale of 2), look from a to k; for the nearest 10 links, from a to c; and for the odd links, look downwards from c to f; thus: EXAMPLE. If it be required to measure off 4 chains 68 links (4.68 chains), to the scale of 2 chains to the inch; look for the first figure of the links (6) at the top of the diagonal divisions, then pass the eye down the diagonal line, and observe where it meets the horizontal line opposite the other figure 8; place one leg of the compasses on this point, and open them till the other leg extends to where the perpendicular line drawn from the 4 chains at the top intersects the same horizontal line. This will be the distance required.* The protractor is also generally furnished with a scale of chords, offering another method of measuring angles. This scale is constructed on the principle, that a chord of 60 degrees equals the radius of the circle; the mode of using this will be shown in Problem 3 in Geometrical Problems. *Note. This distance is marked in the diagram by two small circles (0) at the two points answering to the given distance. |