USE OF THE PRISMATIC COMPASS. 61 may be said of larger instruments, which divide to minutes, and not to seconds. The reader must ascertain for himself the value and capabilities of the instrument, and thereby judge of the extent to which it may be used; much, also, must inevitably depend on the manner; a person with a steady hand will manage this instrument better than another; much, also, depends on the diameter of the instrument used. In Fig. 49, let AB and CD be portions of two main chain lines of a survey; it is required to fill in the fencing along a, b, c, d, e, f; from the point a take the bearing of A B, and of a f, f, being a station on the line CD, and previously fixed for this purpose; fix on some object, in the fence, for instance, near d, in line with af; take also the bearing of a b, and chain to b, offsetting as you go along, and at b take the bearing of ba; if ab bear 125° East, then ba shall bear 125° West; if there is a difference of a few minutes between the readings, take the mean for the correct reading; at b and at c repeat the same work; on reaching d, find this station on the line af, and take the bearings of da and a f; whatever the degree the one is West, the other will be East; in the four-sided figure abcd, the four angles will be equal to four right angles, or 360°; and as, having the bearings, we can obtain the angles from them, we can ascertain how far our bearings give us these four right angles; if, as there is sure to be with this instrument, we find a difference of some few minutes, being over or under the four right angles, divide this difference over the bearings. It is also quite possible that from d we may be able to get a bearing on some known point on the line A B, by which, when we come to plot, we may check the position of b, for if dg reads 130° West, then plotting from hon the plan a line bearing 130° East, will run into station d, when this also has been plotted. For the continuation of our work d, e, f, we have only a triangle, the three internal angles of which should be equal to two right angles, by which we may check the angles at de, f. It cannot fail to be observed that all these bearings may and should be plotted from the station a without once moving the protractor; and as we consider it very desirable that the student should make himself thoroughly master of the prismatic compass before he engages with the principal surveying instrument, the theodolite, we will go a little further into this matter, premising, however, that it will be more as an introduction than anything else to what is termed surveying by traverse, to which we shall come at a future page; probably, also, it will make the subject more familiar to the reader when he comes to it. At a let the bearing of a d be 91° 5' East, and the bearing of ab 136°; note this at the commencement of your line ab before you commence chaining; on reaching the point b, take the bearing of ba, and let this be 136° 5', note this at the end of the line, and take 136° 2' for the mean or corrected bearing East and West of the line, ab; in the same manner, let the mean bearing of b c equal 89°, and that of cd equal 30°; at d take the bearing of da, and let this be 91° 10′; then the mean or corrected bearing of a d or df equal to 91° 7'. Now the angle da b, equal to 136° 24', minus 91° 7′, equal therefore to 44° 55', the angle abc is made up of the angle ab N plus the angle Nbc; and a b N equal to Sab; then a b N equal to 180° minus 136° 24', equal to 43° 57'; and Nbc equal to 89°, the angle a b c equal to 43° 57' plus 89°, equal to 132° 57'. The angle b c d is made up of the angles bc N and Ned, and because N 6S and NcS, the two magnetic meridians, are parallel, and Nbc equal to 89°, then b c N equal to 180° minus 89° equal to 91°, and N cd equal to 30°; then b c d equal to 121°, and cd a equal to Sda minus Sdc equal to N cd equal to 30°, and e da equal 91° 7,* then c d a equal to 61° 7'. And 44° 55′ +132° 57′+ 121° + 61° 7′ = 360°. In the same manner we may check the triangle dcf, the three angles of a triangle being equal to two right angles. Observe that three more lines for filling in have been run from g, on AB to h on CD, and that a tie line has been chained from d to i, being a further check upon the work, for if in plotting d or i lean one way or the other out of position, then the line di will not plot true. For thus filling in, the large 5-inch prismatic compass with silver ring is very useful, and when once the surveyor has become familiar and handy with the use of the instrument it will give very close approximations. To plot these bearings, place your protractor at a, so that A B shall have its due bearing with regard to the magnetic meridian NS, the North and South being each represented on the protractor by 180°; prick off all the bearings, as af, ab, bc, cd, &c., draw af and ab, making the latter the length of the measured chain line; lay your parallel ruler from a to the point pricked off, as the bearing of bc; parallel to this rule in be make b c. With due attention this instrument may also be used for a road survey, as at Fig. 50; the bearings along the bends of the road are taken at the same time the lengths are chained, care being taken at the same time to get bearings from several stations to one or two objects likely to be visible from different parts of Because a straight line meeting two parallel lines makes the alternate angles equal, then S do equal to Nad. USE OF THE PRISMATIC COMPASS. 63 the survey. The work is plotted as before, by laying off the bearings all from one point. Fig. 51 illustrates a similar method of getting a plan of a stream; but here a visual line, A B C, has been set off, for which purpose any two objects are selected to lay down the line; as the bearings of the chain lines along the banks of the stream are measured, a few bearings are taken here and there at different stations upon the two prominent objects, and, if convenient, one or two back sights upon the starting point; if the chain lines have been correctly measured, and the bearings correctly taken, these observations upon the prominent objects will intersect and prove the degree of accuracy of the work, for it must be remembered we cannot expect the same accuracy as with a theodolite; if, however, the prismatic compass is applied in the manner we have been pointing out, errors will be inappreciable. In both Figures 41 and 42 the measurement of the bases A B and A C will completely check the work. There is yet another service to which the prismatic compass is well adapted, which is, from the bearings to ascertain the angles which fences make with chain lines, more particularly those straight fences which do not require offsetting. If in the course of filling in, such a fence be intersected by two chain lines, the intersections of course fix the direction of the fence upon the plan; a check upon the dependence on the chaining only is afforded by the use of the prismatic compass in the manner here mentioned. It may also be rendered very useful in testing the accuracy of a plan the correctness of which we have any reason to doubt. By drawing two or three lines across the plan, and marking on it the angles they make with each other, and then in the field with the plan set out such lines with rods, take their bearings, and see if the angles agree with those measured upon the plan; this may also be done without rods by drawing any line on the plan through two or three junctions of fences, setting off such line accordingly on the ground, and taking the angles which adjacent fences make with the above line, and then comparing these angles with those similarly made on the paper. Supposing that the reader has had sufficient confidence in our advice to follow it with regard to the two former practical lessons we recommended, and that he has satisfied himself of his ability on those points, we now further advise him to practise what we have just observed in this section on the prismatic compass; at first let him content himself with laying out triangles and taking the bearings of the sides; then lay out a series of these three-sided figures, take all the angles, remembering that the three angles of a triangle are equal to two right angles, and let him see how near he can bring this out in practice; let him also chain his lines, and plot the work. In the next place, lay out four and five-sided figures, and so proceed gradually; he may also, by degrees, extend the length of the lines circumscribing the above figures. By this means he will not only make himself perfectly familiar with the use of this instrument, but he will ascertain of his own knowledge its capabilities, and to what extent he may trust himself to use it. This is only to be acquired by one's own practice; and this being acquired, we know what reliance we may place on our own exertions. Before proceeding to a description of the box sextant and the theodolite, we must make a few observations on a particular member of these instruments, which is the following. THE VERNIER. In our most complete and perfect instruments for measuring angles, the limb is divided into degrees, and these are again subdivided into parts of a degree; the number of these subdivisions depends on the size of the instrument, and the degree of accuracy to be desired in the instruments; in some the degrees are halved, which gives a division to 30 minutes; in others the degree is trisected, giving 20', or divided into four, giving 15′, whilst in others the subdivision extends to 10'. Suppose, now, that the index of a theodolite, which corresponds to a visual line, should point to an angle between 21° 30′, and 22°; whatever the difference may be, we can only give a guess at it; this would be very unsatisfactory with an instrument professed almost to attain perfection; to supply this defect an additional scale, called "a vernier," has been contrived to measure parts of space otherwise undefined between any equidistant divisions of a graduated measure. The length of this vernier scale is exactly equal to a certain number of the divisions on the limb, and the number of divisions graduated on the vernier is one more than the number upon the same length on the limb. The length taken on the limb for that of the vernier depends on the degree of minuteness of subdivision to be attained; whatever this may be, the difference of a division upon the vernier from a division on the limb is equal to the nth part of a division upon the limb, n representing the number of divisions on the vernier. If we take two straight lines, or arcs, of exactly equal dimensions, calling one A and the other B, and let one be divided into one more equal parts than the other, then the difference of any two of the equal parts of the two lines will be a fraction, the numerator of which is the length common to both lines, and the denominator the product of the numbers of parts into which each is divided. If we put A for the length of each of the equal lines, as also n and n + 1 for the numbers into which each line is respectively divided, the length of the divisions of each will be, A n A n n + 1, n × (n + 1) In figures let A = 14° 30′, or 29 half degrees, or 870'; then, Suppose, in the limb of an instrument, each degree is subdivided into three parts, or to 20'; it is required that the vernier shall read to 1'; then an arc of 19 parts on the limb divided into 20 parts on the vernier, will give the scale required; the 19 parts on the limb divided to 20', will be 6° 20′, or 380′; then This difference becomes the index for subdividing the smallest division on the limb, and the amount of this subdivision is ascertained by the coincidence of a stroke on the vernier beyond zero, with a stroke on the limb beyond the division last past over by the zero of the vernier; if this coinciding stroke on the vernier be the fifth, then it will be 5', if the ninth, it will be 9', &c., which must be added to the number of degrees, or of degrees and minutes passed over by the zero of the vernier on the limb. As there can be no error as to the point of coincidence of a stroke on the vernier with one on the limb, there can be no mistake as to how many minutes will have to be added, as the measure of a fractional portion of the division on the limb that the zero of the vernier has entered into. A rough judgment will easily be formed as to whether the zero of the vernier has passed over a ora of a division on the limb, and this will readily guide the eye as to the part of the vernier where we must look for the coincidence of one of its strokes with one on the limb; should there, however, happen to be any uncertainty about the exactitude of coincidence, then 20′′, 30′′, or 40′′, may be taken F |