mation to the solution sought for; but if it be not found to be so, the same process must be repeated to a fifth or possibly a sixth pair of points. In carrying out these processes it must be remembered that each new estimate of x may just as probably as not overstep the mark aimed at-that is, lie on the opposite side of the intersection of the curves to that on which lay the last Method of taken x. Thus the successive x's taken may be alternately greater and less or vice versa. procedure Similar general methods of procedure should be followed. when the curves chosen to represent the equations are polar or focal. In these cases the drawing of tangents through the points plotted is even more useful than when rectangular co-ordinates are used provided they can be obtained without much labour. 28. Sometimes special graphic methods of trial and error are much more direct and rapid than the above general method. A good example is furnished by the equation A sin (20 + a) = B sin (0 + ß). Draw two concentric circles of radii A and B (see Fig. 26). Draw any datum radius PO. From PO set off negatively, i.e. to left hand, the angles a = OPa and B = OPb. If either a or ẞ be negative it must be set off to right hand of P O, i.e. in positive direction. Then take any trial point 1 on circle b. With this as centre, and with radius 10 strike the arc O 1′, cutting circle a in 1'. Then if angle O P 1 were 0, angle O P 1', would be 20; a P 1' would be (20+ a) and b P1 would be (0 + B). Also the perpendicular distance of 1' from a P would be A sin (20 + «), and the perpendicular distance of 1 from P would be B sin (0 + B). By trial with the dividers we find the first of these distances much greater than the second. Take a second trial point 2 and find the corresponding point 2' by striking an arc from 2 with radius 2 0. We now find distance of 2' from a P less than distance of 2 from b P. Assume a third point, 3, and find the corresponding point 3'. We find 3' more distant from a P than 3 is from b P. The fourth point, 4, chosen makes these distances equal, and, therefore, OP 4 is the angle e satisfying the equation. In a very short time this angle can be found with a degree of accuracy such that the possible error is less than can be measured on the paper, the error, therefore, being in inverse proportion to the scale of the diagram drawn. Protractors Plotting angles CHAPTER V. GRAPHO-TRIGONOMETRY. 1. We will deal here only with plane trigonometry. We have to make calculations regarding plane figures bounded by straight lines. In doing so we must frequently plot off angles. The instruments called 'protractors' are nearly all of them very rough devices at the best, and are far too untrustworthy for accurate work. The vernier protractor made by Stanley with a silvered divided circle and two opposite arms is a reliable and accurate instrument, but it is costly. The cardboard protractor of 12 in. diameter made by the same maker is also useful, although not so reliable as the other. But as any angle can be set off very easily with ordinary instruments with any desired degree of accuracy, the use of protractors is best wholly avoided. 2. The method is the following, and requires the use of a table of chords, such as one finds in Chambers's or in most other mathematical tables. The table found in Molesworth is not sufficient for the plotting of angles taken in surveys, because it gives the chords for every degree only; whereas the angle is read in the field to minutes. First draw from the centre from which the angle is to be plotted a circle of, say, 10 in. radius. On this circle the chord of the desired angle is evidently 10 in. multiplied by the tabulated chord for the given angle. This quantity in inches, taken in the dividers, is set off as the chord of the desired angle. For example, suppose the angle to be plotted is 69° 22′. We find 10 chord 69° 22' = 11.38, and set this off as the chord on a circle of 10 in. radius. Now, from the table we find that 11.39 = 10 chord 69° 27′ and 11.37 = 10 chord 69° 17'. Now, it is possible with ordinary care to set the compasses to in., but much greater accuracy than this is not easily possible. Thus we cannot pretend to set off very accurately as a chord anything between 11.37 and 11.38 or between 11.38 and 11.39. These, as we have seen, correspond on a circle of 10 in. radius to angles differing by 0° 5'. With this size of circle, then, we cannot pretend to plot angles to any greater accuracy than 5'. With very small angles, indeed, the accuracy is increased to 3', but with angles larger than the above it is considerably Plotting reduced. Thus it is advisable never to plot off by this method angles angles greater than 45°. The complement of the angle-for instance, 90° - 69° 22′ = 20° 38'-should be plotted off instead. This is also the degree of accuracy obtainable with a circular protractor without vernier, with a divided circle of 20 in. diameter. If a circle of 20 in. radius is used, an accuracy of 2' is obtainable in plotting. To make the construction on this large scale requires beam-compasses, and, of course, to maintain this accuracy throughout the diagram it requires to be drawn to a correspondingly large scale. 3. The solution' of any triangle or other plane rectilinear figure is accomplished graphically by plotting it off accurately to scale, and measuring the quantities desired. If it is a Triangles length that is to be found, it is measured in the ordinary way; rectian angle is to be measured by a reversal of the above ex- figures and linear plained process; and the measurement of areas we will immediately proceed to explain. But it should first be observed that, in plotting off, all the angles required should be set off in the first place upon one and the same circle, and the directions so obtained then transferred by sliding set squares upon straight-edges into the positions required in the drawing. That is, we are not to draw a new 10 in. circle at each new Datum point of the drawing where an angle is to be set off, because such a proceeding would involve the waste of time and labour, circle because it would cover the paper with unnecessary and confusing lines, and because at several of these points it will usually be found that there is not room inside the limits of the paper to use a good-sized circle. The centre of the marking-off circle is conveniently chosen near the middle of the paper. Thus in Fig. 27 let it be required to mark off from the line OA downwards the angle 140°. Instead of doing this directly, we mark off upwards 180° — 140° = 40° from A to B, and thus get B O as the desired direction, which can now be transferred to any part of the drawing. Again, let it be required to mark off 70° downwards from OA, the angle A O V1 being less than 70°. We first mark off 20° to O C1 from OHLO V, then take the chord A C1 in the dividers and set it off from V to C. We thus obtain C O as the direction wanted. We now give two examples of plotting off. In Fig. 28 is shown the calculation of the height of a church spire from Height of theodolite measurements. The theodolite is first placed at a a spire station A. The height of its axis from the ground is measured 4.12 ft. The angle of elevation to top of spire is measured 43° 25'. Another station B is chosen in line with A and top of spire. Distance A B on ground is measured 120 ft. Difference of level of ground at B and A is measured by reading with theodolite at A placed with telescope level on surveyor's levelling staff held at B, 2·34 ft. Theodolite is now placed at B, and height of its axis above ground measured 4.47 ft. The angle of elevation to top of spire is measured 25° 15'. The construction is so simple and so easily understood from the figure that no explanation is needed. The marking off circle is struck from O, the position of the theodolite axis at station B with radius OH = 5 in. The line a' is drawn parallel to O a, the angle H O a being made 43° 25'. From x the intersection of a' with Ob is measured X down to the ground line through station A. This is the height required. The distance Y from station A to centre of base of spire may also be directly measured from the diagram. The calculation |