58. The word Intersection is used when two lines cut each other, and the place where they cross is called the Point of intersection. 59. The word Bisect, signifies to divide into two equal parts, and Trisect to divide into three equal parts. 60. To Produce or Prolong a straight line is to lengthen it in the same straight line. 61. A Generatrix is that by which something is generated; thus, to give motion to a point, it becomes a generatrix, and a line is the result. In like manner a line may be said to generate a plane; and a plane, a solid. In these operations the generatrix or Generatrice is the generating element, and the thing generated is called the Generant. 62. A Directrix or Dirigent, is the line of motion, along which a line or plane, called a Describent, may be conceived to move, to describe a plane or solid. The words describe, construct, make, and draw, are frequently used in practical geometry, in one and the same sense. A Problem is something proposed to be done. A geometrical representation of a building or other solid, when seen vertically, supposing the eye to be stationed at an infinite distance, is called a Plan. If the eye be conceived to be situated at an infinite distance horizontally, the drawing is then called an Elevation. If a solid be conceived to be cut by a plane passing through it in any direction, at right angles to the line of vision; a drawing of the internal parts thus exposed is called a Section. The plural number of radius is radii; of superficies, superficies; of generatrix, generatrices; of directrix, directrices; of vertex, vertices; of rhombus, rhombi; of focus, foci; and of trapezium, trapezia. The plural of other names is made by adding the letter s to the singular; as, tangent, tangents, &c. A degree is divided into sixty equal parts, called minutes; these into sixty equal parts, called seconds; and these also into sixty equal parts, called thirds. Note. Such definitions as have not diagrams annexed, are either illustrated by those of other definitions, or by those of the following Problems. The Definitions and terms being understood, the student may turn his attention to drawing geometrical figures; but to do them neatly and accurately, mathematical instruments well finished are indispensable. To obtain these, application should be made to a respectable maker. Good second-hand instruments are sometimes to be met with at a cheap rate; but the novice should never purchase any, except at the recommendation of a competent judge. If badly filed, or if the points be not tempered steel, they will be of little or no service. The smallest number that can be available, must contain a pair of compasses with shifting leg; pen and pencil legs to fit in the compasses; and an ivory protractor, with scales engraved on it. A larger set contains, besides these, a pair of dividers, a long drawing pen, bow pen, and a bow pencil; the two latter for circles and arcs not exceeding an inch radius. Still larger sets contain hair compasses, lengthening bar, sector, and parallel ruler: these can be dispensed with, particularly the three latter. The sector is seldom used; and a Triangle and Ruler," made by a good carpenter, answer all the purposes of a parallel ruler, and are not so likely as it to get out of repair. 66 Diagrams must not be drawn till the student is a little acquainted with his instruments. He must look at each one separately, and try to discover its use, which he will find little difficulty in doing. He may then draw, with a hard lead pencil, straight lines, and concentric circles, following the edge of the ruler for one, and taking care to press very gently on the compasses in describing the circles, lest the centre should be worn to a large hole. The lines must now be "inked in;" this is to be done with the best indian ink, rubbed up very black, and put into the drawing pens with a camel's hair pencil. When the ink is dry, the lines may be rubbed gently with indian rubber, to remove the lead pencil, and each line examined to ascertain whether the whole are equal, clear, and of uniform thickness; for to draw a good line is a great desideratum, and is moreover not an easy thing to do. When continued lines of various thicknesses can be drawn at pleasure, dotted lines may be undertaken, both right and curved; but those lines intended to be dotted in ink, are not to be first dotted with lead pencil. Specimens of dotted lines may be seen in the diagrams accompanying the problems. It will likewise be useful to draw a line two or three inches long, and divide it, by repeated trials, into two, three, or four equal parts, so that the eye may become accustomed to judge of equal distances; for although Problems are provided for such cases, yet instances often occur in which divisions sufficiently accurate can be made by this method, (with the Dividers,) and in much less time than by any other process. A Problem in Practical Geometry supposes three things; first, something is given; secondly, something is wanted; and thirdly, to obtain the second from the first, certain means must be employed; these are called lines of construction, and are always to be dotted. Dotted lines are employed for three purposes; first, to show the shape of those parts in solids that are hidden by some opaque covering. For example, if the cylinder of a steam engine be drawn, although the piston cannot be seen through the iron work, yet its shape can be accurately described by dotted lines, without interfering with the truth of the other parts. Secondly, dotted lines are used in machinery, to show the direction of motion; thus, if an artist had drawn the beam of a steam engine, and then wished to illustrate its motion, he would show the beam in its next proper position by dotted lines. Thirdly, as already stated, lines of construction in problems must be dotted. These are supposed to be rubbed out after the problem is completed, being no longer necessary. They may be compared to a scaffold, which is to be removed when the building is finished. They are, however, suffered to remain dotted in geometrical figures, to guide the student. Note 1. The lines forming the figures should be drawn in ink, in the same order as they were done in pencil, to fix the process of construction in the student's memory. Note 2. Where dimensions are given in inches, in the Examples for Practice, feet, or other magnitudes may be substituted if wanted. PROBLEMS. PROBLEM I. To bisect a given rectilineal angle. 1. Draw two right lines of any length, and let them contain any angle. Call this the given angle. Print or write (with the lead pencil) any letters at the extremities of these lines, (as BA C, in the diagram annexed,) simply that the lines may be easily referred to. 2. Having fixed the lead pencil leg in the compasses, place the steel leg very accurately on the vertex A. Gently open the compasses any distance less than the length of the line A B, or A C. Call this B D distance a radius, and with it describe an arc, as at D E, (with the pencil leg,) cutting the lines A B and A C, in the points D and E. Print the letters D, E. 3. Next, place the steel leg carefully on the point D, and with the same radius as before, (or any other, greater than half the distance from D to E,) describe an arc, (with the pencil leg,) as at F. Remove the steel leg to the point E; and with the radius last used, describe another arc, as at F, cutting the first arc in the point F. 4. Draw the right line A F, and it shall bisect_the_angle BAC, as was required to be done; for the angle B AF shall be equal to the angle CA F. To test the truth of the work in this problem, apply the dividers from D to the point of intersection of the bisecting line with the arc; and if the distance agree with that from E to the same point of intersection, the diagram is correct. The process explained in this problem, will, with equal accuracy, bisect a curvilineal angle, provided its legs are curved equally. When the diagrams are drawn in ink, the pencil lines must be rubbed out. The letters of reference may also be rubbed out, or inked in, at pleasure. EXAMPLES FOR PRACTICE. 1. Draw an acute angle, and bisect it. 2. Draw an obtuse angle, and bisect it. PROBLEM II. At a given point in a given right line, to make an angle equal to a given rectilineal angle. 1. Let ECD be the given angle; A B the given right line; and A the given point. It is required from A to draw a line GA, which, with B A, shall contain an angle equal to the angle E CD. 2. From the vertex C, (in the given angle,) as a centre, and with any radius, describe an arc, cutting the legs of the given angle in the points E and D. 3. From the point A (in the given right line) as a centre, and with the radius CE, describe an arc G F, cutting AB in the point F. E Di W A FB 4. Take D E as a radius, and from the point F as a centre, describe an arc, cutting F G in the point G. 5. Draw the right line A G, when the angle G A B shall be equal to the angle ECD. EXAMPLE. Make any obtuse angle, and make another angle equal to it. PROBLEM III. To make an angle that shall contain any given number of degrees. 1. Draw a line as A B in Problem II. 2. Take 60 degrees in the compasses, from any line or scale of chords (marked CHO or C) on an ivory scale, and from A as a centre, describe the arc G F. 3. Take the number of degrees which the angle required is to contain, from the same line of chords; and from F as a centre, describe an arc, cutting G F in G. 4. Draw A G, and G AB will be the angle required. |