the figures may be completed in each case, and remembering that the hexagon is the most common of all plane figures drawn by the mechanical engineer, as nuts and bolt heads are hexagonal, it is very necessary for the student to know to what extent set squares may be used to assist in its construction. EXAMPLES. EX. 16.-Draw two lines 2′′ long meeting at an angle of 108°, and consider them as two sides of a regular polygon. Complete the figure. EX. 17.-Construct the following regular polygons:-(a) pentagon, 2" side; (b) hexagon, 2′′ side; (c) heptagon, 1.75′′ side; (d) octagon, 1.5′′ side; (e) nonagon, 1.5′′ side. EX. 18.-Draw a line A B, and take a point, P, outside it, 31" away. Construct a pentagon to have one side in A B, and the opposite corner in P. (Construct any pentagon, then copy this pentagon by parallels having its top corner in P, and mark the side opposite P, CD. Join P through C and D to meet A B, the length they cut off on AB is the side of the required pentagon, then finish by drawing parallels.) EX. 19. Construct the regular polygon whose perimeter is 10", and interior angles 135°. (To do this by construction, a polygon of any length of side having interior angles of 135° should be drawn first, as this will tell the number of sides, then draw a similar polygon such that perimeter 10".) = The following should be drawn, using set squares :— EX. 20.-Draw a hexagon 2′′ side and an octagon 13" side. EX. 21.-Draw a hexagon and octagon, outside and inside circles of 3" diameter. EX. 22.-The longest diagonal of a hexagon is 4′ and of an octagon 5". Draw the figures. EX. 23. The distance between the parallel side of a hexagon is 44", and of an octagon 5". Draw the figures. PROBLEM XII (Fig. 18).-To construct a regular polygon inside a given circle. Let the circle have the diameter, A B, and the required polygon be a heptagon. Divide the diameter of the given circle, A B, into the same number of equal parts as the sides in the required polygon; for a heptagon 7, and mark as shown. With A and B, as centre, radius A B, describe arcs meeting in C. Join C through the point 2 to meet the circle on the other side of the diameter A B in the point D. Then the line A D is one side of = the heptagon, and the figure can be completed by stepping off the length A D around the circle. EX, 24.-Construct an equilateral triangle, ABC, base AB, divide the base into 5 equal parts, and on the side of the base remote from C describe a semicircle. Draw a line from C through the second division point 2 counting from A, and meeting the semicircle in D. Measure the angles, B 2 D and A 2 D, and the chords, BD and A D, and show that EX. 25.-Construct the following polygons in circles of 4" diameter (a) pentagon, (b) hexagon, (c) heptagon, (d) octagon, (e) nonagon. EX. 26.-Construct two hexagons having the same centre, length of sides 21", the sides of one hexagon to make an angle of 30° with the sides of the other. Construction of Ellipses.-The ellipse is the most common of a series of useful mathematical curves, often employed in architectural and engineering construction, many of which will be referred to in detail in Section VII. But its geometrical construction is given at this stage, because of its occurrence in other work, and the desirability that students should obtain an early knowledge of how it may be practically drawn. The general method of construction adopted with all thesecurves, is to find a number of points through which it is known the curve must pass, and then to draw, by freehand or with the aid of "French curves," the curve passing through these points. Arcs of circles cannot be used with any degree of accuracy. The greater the number of points found, the more accurate the curve is likely to be, but the student should learn to exercise a wise discretion as to the exact number of points in particular cases. As the curves are symmetrical, any error in drawing is easily detected. The ellipse may be defined in many ways, but for the present we will take the following definition : "An ellipse is a closed curve traced out by a point moving in such a way, that the sum of its distances from two fixed points, called the foci, is always the same." Thus, in Fig. 19, if F and F' are the two fixed points or foci, and P the moving point, then if P moves so that at all times PF + PF = a constant, then the path of P is an ellipse. Major Axis.-The line passing through the foci, and terminated by the curve, is called the "major axis" (AB in Fig. 19). Minor Axis. The line bisecting the major axis at right angles to it, and terminated by the curve, is called the "minor axis" (O D in Fig. 19). The intersection of the axes is called the centre of the ellipse (O in Fig. 19). Ordinates.-Lines parallel to the minor axis and terminated by the curve are called "ordinates." = = Since A and C (Fig. 19) are points in the ellipse, it follows that AF+AF' CF CF; but A F + A F major axis, A B; therefore, as C F and C F' are equal, we have C F half major axis. Therefore, = The sum of the distances of any point in an ellipse from the foci is equal to the major axis. The distance from either end of the minor axis, to either focus, is equal to half the major axis. A circle may be regarded as an ellipse with its axes equal, and a straight line as an ellipse with its minor axis infinitely reduced. There are several means of constructing an ellipse when the axes are known (or one axis and the foci, since the other axis is then easily found), the first of which is suggested by the above definition. PROBLEM XIII. (Fig. 19).—To construct an ellipse when the axes A B and C D are given. Method I. by " arcs of circles."-Find the foci, F and F', by taking half the major axis as distance, either end of the minor * D Fig. 19. axis as centre, and cutting the major axis in these points. Mark any point 2 in the major axis A B, between the foci. With A 2 as distance, F as centre, draw an arc. With B2 as distance, F' as centre, draw an arc, cutting the first arc in P. Then PF + PF' a point in the ellipse. = A 2+ B2 = major axis; therefore P is In the same way by taking other points, 1, 3, &c., additional points can be found, and the curve drawn through them. The points may be taken anywhere between the focus and centre, but are better when closest together nearest the focus. Arcs can be drawn with the same radii on both sides of the major axis, and with both foci as centre, thus giving four points in the curve for each of the points, 1, 2, 3, This is a quick and accurate way of constructing an ellipse. It can also be applied for constructing other curves of a similar character, such as, for instance, where 2 PF + PF': = a con stant. It is evident that the curve could be drawn mechanically. For let FP, P F' be a continuous string, its ends being fixed at the foci. Then a pencil guided by the string, and keeping it tight, will describe an ellipse. Method II. by "two circles" (Fig. 20).-Draw circles with centre, O, having the major and minor axes for diameter. Take any point, 3, on the major axis circle, and join to the centre, cutting the minor axis circle in point 4. Through the point (3) on the major axis circle, draw a line parallel to the minor axis. Through the point (4) on the minor axis circle, draw a line parallel to the major axis, meeting the first line in e. Then e is a point in the curve. Repeating the construction with other points will enable the ellipse to be drawn. Since the curve is symmetrical about its axes, points in it can be found, when one quarter has been constructed, by drawing ordinates from points in that quarter, and making them of equal length on both sides of the major axis. Thus, in the figure ef=fg, and similarly by drawing lines parallel to the major axis, so that mn = gm, the ellipse can be completed. It is better to draw one complete half of the ellipse before adopting this method. Method III. by an "oblong" (Fig. 21).-Draw an oblong, ABCD, having the axes for diameters. Divide half the major axis, A O, into any number; say, six equal parts and mark from A towards O, 1', 2', 5'. Divide the distance, AE (equals half minor axis), into the same number of equal parts, and mark from A towards E, 1, 2, 3, .. 5. Join these points to the end, C, of the minor axis. |