H C GH at G, the middle of the line EF; join HE, and it will be equal to HF. Now the straight line CE is shorter than the bent line CHE, or than its equal CH F. Conversely the longest oblique has the greater departure. Let CF be greater than CE, then F is further from the perpendicular than E; were nearer, CE would be less than C F; same distance, then C E and C F would be one another; therefore F is further from the perpendicular than E. G E D B Fig. 34. for if it if at the equal to From a point to a given straight line, there cannot be drawn more than two equal straight lines. 40. For a third line would be more or less remote from the perpendicular than either of the first two, and consequently would be greater or less than it. This equality of oblique lines, equidistant from the foot of the perpendicular, offers a very convenient means of testing perpendiculars. The mason's level. 41. Take a board (fig. 35) with one of its sides, A B, well planed; upon this side raise a perpendicular, which we will call the square-line; from a point in it fix a plumb-line, so that the weight may oscillate in a hollow made in the board: the Fig. 35. whole forms an instrument termed a mason's level. Place the board upright; if the lower edge be perfectly horizontal, the plumb-line, being vertical, will cover the square-line. In fig. 36 the mason's level is used to see whether a stake driven into the ground at B, is level with the stake A. The figure shows that this result has not been obtained, for the plumb-line does not fall upon the perpendicular; and as it is to the left of this line, the stake B, to the right, must be lowered still farther. In using the level, care must be taken to incline it to the front from time to time, in order that the line may become steady. To ascertain whether a stone is level, place the instrument upon it in different positions; if the plumb-line fall invariably upon the square-line, the surface of the stone is perfectly horizontal; if not, the stone must be wedged in such a manner that the desired result may be obtained. The instrument must be often tested and corrected, for the wood wears unequally. It may be tested and corrected simultaneously by dropping a new perpendicular from the fixed point to the base; or, having driven in two stakes (fig. 36), until the plumb-line lies on the marked line, turn the level round so that the end which was on A is on B. If the instrument be true it will give the same indication as before.. To draw an angle equal to a given angle. 42. An instrument called a false-square, or bevel (fig. 37), is used for this purpose. It consists of a blade and stock, similar to the parts of the carpenters' square, which turn upon an axis, like a pair of compasses, so as to form an angle of any size. The interior or exterior edge of the stock is applied to one of the sides of the given angle, and the blade is opened or shut until its edge lies upon the other. The instrument is then taken up, and an angle equal to the given angle may be drawn in any position. Joiners frequently use this instrument; as, for example, when they are required to fix a shelf in a corner cupboard, the angle formed by the two sides of the cupboard is first measured by the instrument, and the shelf cut to fit the angle thus obtained. Questions for Examination. 1. What is meant by a vertical line, and what by a horizontal line? 2. Define a plane. Show how to test the evenness of a drawing-board. 3. Define a straight line. of a ruler. Show how to test the straight edge 4. Show how to measure a straight line on land. 5. Explain the terms angle, right-angle, acute angle, obtuse angle, perpendicular, and oblique. 6. Prove that only one perpendicular can be drawn from a point in a straight line. 7. Prove that the sum of all the angles formed by a number of straight lines that meet in the same point is equal to four right-angles. 8. Prove that when two straight lines cut one another the vertically opposite angles are equal. 9. Show how to form a square angle. Show how to test a setsquare. 10. Explain how a perpendicular may be drawn with the set-square. 11. What is meant by by a "geometrical locus." Prove that the locus of points equally distant from the extremities of a straight line is the perpendicular to its middle point. 12. What are the requisites of a T square? How is it used? 13. Prove that every oblique line from a point to a straight line is greater than the perpendicular from the same. 14. Equal oblique lines from a point to a straight line are equally distant from the perpendicular: show how to employ this property in testing perpendiculars. 15. Describe the mason's level and the method of testing it. Theorems and Problems. 1. If two straight lines intersect, and one of the angles formed is a right angle, so are the other three. 2. The bisectors of the two adjacent angles which one straight line makes with another are at right-angles. 3. The bisectors of two vertically opposite angles are in the same straight line. 4. From a given point on a straight line, two straight lines and only two can be drawn, making with it a given angle. 5. One and only one perpendicular can be drawn from a point to a straight line. 6. How would you draw a perpendicular to the side of a field by means of a cord? 7. There are two roads connecting two villages, A and B, one along a bent line, A C, C B, and the other along a bent line, A D, DB. The point D lies with the angle A CB; prove that the road A C B is greater than A D B. 8. Two villages, A and B, are connected by straight roads with two others, C and D. The roads A D and C B intersect, but A C and D B do not. Prove that the roads which intersect are together longer than those which do not. 9. Light travelling from A falls upon a mirror, EF, at B, and is reflected to C in such a way that the angle A B E is equal to CBF. Prove that the path of the light is shorter than it would be if, being reflected at any other point in the mirror, it could pass from A to C. 10. A and B on opposite sides of a line E F are joined with a point C in EF. Prove that the difference between A C and B C is greatest when the angles A CE, BC E are equal. II. Show how to draw from a given point a straight line which shall make equal angles with two given straight lines. Arithmetical Questions. 1. How many standard yards are equal in links? 2. How many links will be equal in length to length to 2761 Ans. 607.42. 1760 feet? Ans. 2666. 3. Express a length measuring 387.53 chains in yards, feet, and inches. Ans. 8525 yds. I ft. 11.76 in. 4. How many links are equal in length to 17,776 inches, and how many to 9 miles? Ans. 22444; 72,000. CHAPTER III. Circumferences.-Use of compasses. 43. A circle is a figure contained by one line, termed the circumference, such that all straight lines drawn from a certain point within the figure to the circumference are equal. If we place one of the points of a pair of compasses upon a perfectly even board, and the compasses be then turned round this point, the other being properly pressed upon the board, a line is drawn which we call a circumference (fig. 38). The distance between the two points of the compass is called the radius of the circle, the fixed point being I called the centre. It is easily seen Fig. 38. that the straight lines which join different points in the circumference with the centre are all of the same length; that is, all radii in the same circle are equal. Two circumferences having the same radius are equal. 44. With the compass opened to the same extent, |